260
Magnetic order, spin-induced ferroelectricity and structural studies in multiferroic Mn 1´x Co x WO 4 by I RENE URCELAY OLABARRIA THESIS SUBMITTED TO THE DEPARTAMENT DE FÍSICA OF THE FACULTAT DE CIÈNCES OF THE UNIVERSITAT AUTÒNOMA DE BARCELONA FOR THE DEGREE OF DOCTOR OF PHILOSOPHY c a -4 -3 -2 -1 0 1 2 3 4 -1 0 1 -4 -2 0 2 4 l k h T = 2 K ( ω t ) c a P ( ω t ) - 4 - 3 - 2 - 1 0 1 2 3 4 - 1 0 - 4 - 2 0 2 4 l k k k k k k h T=2K 1 T = 2 K K K K 1 1 1 m m l lj = k k k k k k k k k k k k k k k k k k k k k k k k k k k k k S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S S k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k j j j j j j j j j j j j j j j j j e e e e e e e e e e e e e e e e e e e e e e e e e e e e e 2 π π π π π π π π π π π π π π π π π π π π π π i i i i i i i i i k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k · · · · · · · · · · · · R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R R l l l l l l l l l l l l l l l l l l l l 0 0 + H H R · r ν ν ν ν ν ν )= ) )= )= )= )= )= = )= )= )= )= )= )= )= )= )= ) )= = )= = )= = = = )= ) = ) ) ) ) ) ) ) ) ) ) ) ) = ) ) ) ) ) ) ) ) ) ) ) = = = ) ) = = ) ) ) ) ) θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ det det det ( ( ( R R R R R R R R R R R R R R R R R R R R R R R R R ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) R R R R R R R R R R R R R R R R · M M M M ν ν ( ( ( ( ( ( 3 x x x x x x x 3 3 4 4 4 4 4 4 4 ) ) ) ) ) 4 4 ) + + H )= p p pf pf pf f f pf pf f pf f 0 0 f ( ( ( ( Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q ) ) ) ) ) ) ) ) ) ) ) ) j S S S S S S S S S S S S S S S S k k k k k k ,j j e e e e e e e e i i i i i i i i i i i i i i Q Q Q Q Q Q Q · r r j j j e e W W j χ χ χ χ = C C C C C C C C C C C C C T Θ Θ Θ Θ P P P P P P P P χ χ χ χ χ χ χ χ χ χ χ χ χ ac ac a a ac ac ac ac a ac ac = = = = = = = = = = = = = = χ χ χ cos( s( s( s( ( ( ( ( ω ω ω ω ω t t t t )+ )+ )+ )+ )+ )+ F F F F F F F F F F F F F F F F F M M M M M M M M M M M M M M M M M ( ( ( ( ( ( ( ( ( ( ( ( ( Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q = = = = = = = = = = k k k k k k k k k k k ω ω ω ω ω ω t t t t + + + + + + 1.2 1.4 1.6 T N T N2 a c χ (10 -4 CGS) b H = 1 kOe 10 12 14 1.45 1.50 1.55 c axis T N2 T N s s s si s n( P r ij j j j j j j ij ij × × × × × × × × × × × × × × × × × × × × ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( S S S S S S S S S S S S S S S S S S S S S S S S S S S i i i i i i i i i i i i i i i i i i i i i × × × × × × × × S S S S S S S S S j j j j j j j j j S S S S S ) ) ) ) ) ) ) ) ) ) n( M M M M M M M M M M M M M M M M M μ μ μ μ μ μ μ μ μ μ μ μ M M M M ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( R R R R R R R R I I I I x x x 4 4 4 4 + τ 0 τ τ )+ )+ )+ χ χ χ χ χ s s si s s s s n ω ω ω ω t t t t t )+ )+ )+ )+ + + )+ )+ χ χ χ 1 . 2 0 5 10 15 20 0 25 50 75 100 P b P c P a P (μC/m 2 ) Temperature (K) 5 10 15 1.00 1.01 1.02 1.03 ε c x10 ε a Temperature (K) ε(T)/ε(5K) SUPERVISORS: DR.VASSIL S KUMRYEV DR.ERIC RESSOUCHE DR.J OSE LUIS GARCÍA MU ˆ NOZ December, 2012

Magnetic order, spin-induced ferroelectricity and ... · Magnetic order, spin-induced ferroelectricity and structural studies in multiferroic Mn 1´xCo xWO 4 by IRENE URCELAY OLABARRIA

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Page 1: Magnetic order, spin-induced ferroelectricity and ... · Magnetic order, spin-induced ferroelectricity and structural studies in multiferroic Mn 1´xCo xWO 4 by IRENE URCELAY OLABARRIA

Magnetic order, spin-induced ferroelectricityand structural studies in multiferroic

Mn1´xCoxWO4

by

IRENE URCELAY OLABARRIA

THESIS SUBMITTED TO THE

DEPARTAMENT DE FÍSICA OF THE FACULTAT DE CIÈNCES OF THE

UNIVERSITAT AUTÒNOMA DE BARCELONA

FOR THE DEGREE OF

DOCTOR OF PHILOSOPHY

c

a

-4 -3 -2 -1 0 1 2 3 4

-1

0

1

-4-2

024

l

k

h

T = 2 K

(ωt)

c

aP

(ωt)

-4 -3 -2 -1 0 1 2 3 4

-1

0

-4-2

024

l

kkkkkk

h

T = 2 K1

T = 2 KKKKKK111

mmllj =∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑kkkkkkkkkkkkkkkkkkkkkkkkkkkkk

SSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkjjjjjjjjjjjjjjjjjjjeeeeeeeeeeeeeeeeeeeeeeeeeeeee−2ππππππππππππππππππππππππiiiiiiiiiikkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkk·············RRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRRlllllllllllllllllllll

τ00ττ+HHR

· rνννννν)=)) =) =) =) =) =) =) =) =) =) =) =) =) =) =) =)) ==) ==) ====) =) =))))))))))))) =))))))))))) ===)) ==)))))θθθθθθθθθθθθθθθθθθθθθdetdet

det((((RRRRRRRRRRRRRRRRRRRRRRRRRR

)))))))))))))))))))))))))))))RRRRRRRRRRRRRRRR·MMMMνν

((((((3xxxxxxx33 4444444)))))44)

++H) = pppfpfpfffpfpffpfff00fff(((((((((QQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQQ))))))))))))))))))

∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑∑jjjjjjjjjjjjj

SSSSSSSSSSSSSSSSkkkkkkkkkk,jjjeeeeeeeeeeeeeiiiiiiiiiiiiiiiiiiiiQQQQQQQQQQQQQQQQQQ··rrrrjjjjeeWWjW

χχχχ =CCCCCCCCCCCCC

T −ΘΘΘΘPPPPPPPP

χχχχχχχχχχχχχacacaaacacacacaacac ============== χχχχ ′cos (s (s (s (((((ωωωωωtttt) +) +) +) +) +) + ′

FFFFFFFFFFFFFFFFFMMMMMMMMMMMMMMMMM(((((((((((((QQQQQQQQQQQQQQQQQ ========== kkkkkkkkkkk

ωωωωωωtttt ++++++1.2

1.4

1.6TN

TN2

acχ

(10-4

CG

S) b

H = 1 kOe

10 12 141.45

1.50

1.55c axis

TN2

TN

ssssis n (

P ∝∝∝ rijjjjjjjijij ×××××××××××××××××××× ((((((((((((((((((((SSSSSSSSSSSSSSSSSSSSSSSSSSSSiiiiiiiiiiiiiiiiiiiii××××××××× SSSSSSSSSjjjjjjjjjSSSSS ))))))))))

n (

MMMMMMMMMMMMMMMMMμμμμμμμμμμμμμμMMMMM((((((((((((((((((RRRRRRRR

IIIIxxx4444

+τ0ττ

) +) +) +++ χχχχχχχ ′′′′′′sssissss n

(((ωωωωωttttt) +) +) +) +++) +) + χχχχ ′′′′′′′′′′′′′1.2

0 5 10 15 200

25

50

75

100

Pb

Pc

Pa

P (μ

C/m

2 )

Temperature (K)

5 10 151.00

1.01

1.02

1.03

εcx10

εa

Temperature (K)

ε(T

)/ε(5

K)

SUPERVISORS:DR. VASSIL SKUMRYEV

DR. ERIC RESSOUCHE

DR. JOSE LUIS GARCÍA MUNOZ

December, 2012

Page 2: Magnetic order, spin-induced ferroelectricity and ... · Magnetic order, spin-induced ferroelectricity and structural studies in multiferroic Mn 1´xCo xWO 4 by IRENE URCELAY OLABARRIA
Page 3: Magnetic order, spin-induced ferroelectricity and ... · Magnetic order, spin-induced ferroelectricity and structural studies in multiferroic Mn 1´xCo xWO 4 by IRENE URCELAY OLABARRIA

SUPERVISORS:DR. VASSIL SKUMRYEV

DR. ERIC RESSOUCHE

DR. JOSE LUIS GARCÍA MUNOZ

December, 2012

Magnetic order, spin-induced ferroelectricityand structural studies in multiferroic

Mn1´xCoxWO4

by

IRENE URCELAY OLABARRIA

THESIS SUBMITTED TO THE

DEPARTAMENT DE FÍSICA OF THE FACULTAT DE CIÈNCES OF THE

UNIVERSITAT AUTÒNOMA DE BARCELONA

FOR THE DEGREE OF

DOCTOR OF PHILOSOPHY

Page 4: Magnetic order, spin-induced ferroelectricity and ... · Magnetic order, spin-induced ferroelectricity and structural studies in multiferroic Mn 1´xCo xWO 4 by IRENE URCELAY OLABARRIA
Page 5: Magnetic order, spin-induced ferroelectricity and ... · Magnetic order, spin-induced ferroelectricity and structural studies in multiferroic Mn 1´xCo xWO 4 by IRENE URCELAY OLABARRIA

Dr. Vassil Skumryev, Dr. Eric Ressouche andDr. José Luis García Muñoz

CERTIFY

that the work reported in the present by Irene Urcelay Olabarria, entitled"Magnetic order, spin-induced ferroelectricity and structural studies in multifer-roic Mn1´xCoxWO4", has been done at the Institut Laue Langevin in Grenoble un-der their supervision. This work constitutes the Doctoral Thesis Memory submit-ted by the interested person to the Facultat de Cièncias de la Universidad Autònomade Barcelona to apply for the degree Doctor of Philosophy.

Bellaterra, 19th of December, 2012.

Signature:

Dr. José Luis García MuñozResearch Professor of the Insti-

tuto de Ciencia de Materiales

de Barcelona, Consejo Superior

de Investigaciones Científicas.

Signature:

Dr. Eric RessoucheResearch Director,

CEA/Grenoble,

INAC/SPSMS.

Signature:

Dr. Vassil SkumryevProfessor of the Institut Català

de Recerca i Estudis Avançats

(ICREA) in the Dep. de Física

of the Universitat Autònoma

de Barcelona.

Page 6: Magnetic order, spin-induced ferroelectricity and ... · Magnetic order, spin-induced ferroelectricity and structural studies in multiferroic Mn 1´xCo xWO 4 by IRENE URCELAY OLABARRIA
Page 7: Magnetic order, spin-induced ferroelectricity and ... · Magnetic order, spin-induced ferroelectricity and structural studies in multiferroic Mn 1´xCo xWO 4 by IRENE URCELAY OLABARRIA

Ama ta aitandako

Page 8: Magnetic order, spin-induced ferroelectricity and ... · Magnetic order, spin-induced ferroelectricity and structural studies in multiferroic Mn 1´xCo xWO 4 by IRENE URCELAY OLABARRIA
Page 9: Magnetic order, spin-induced ferroelectricity and ... · Magnetic order, spin-induced ferroelectricity and structural studies in multiferroic Mn 1´xCo xWO 4 by IRENE URCELAY OLABARRIA

Acknowledgements

Time flies. Actually very fast. It is already more than three years since this storybegan. I have found many people in the path who, in some way or another, helpedand supported me. I would like to thank them all, because without them reachingthis point of the adventure would have been tougher than what it has been, oreven impossible.

First of all, I would like to thank the Institut Laue-Langevin for giving me athree-year fellowship to do the thesis. I would like to thank Prof. J. Rodriguez-Carvajal for kindly accepting me in the Diffraction Group, as well as to all themembers of this group.

This work would not be possible without the hundreds of hours of beamtime used in distinct installations. Therefore, I am grateful to the Institut Laue-Langevin, to the CEA-Grenoble (CRG-D23), to the Spanish CRG-D15, LaboratoireLéon Brillouin and European Synchrotron Radiation Facility for the provision ofbeam time.

I acknowledge the financial support from MICINN (Spanish government)under Project No. MAT2009-09308.

I would like to thank my supervisors, for their help and guide.

I am very grateful to Prof. Alexander Mukhin and his team for providing uswith the samples and the electric measurements as well as for the useful discus-

ix

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sions we have had.

I would like to thank Prof. Arsen Goukasov and Dr. Caroline Curfs for theirhelp in the experiments performed in the Laboratoire Léon Brillouin and EuropeanSynchrotron Radiation Facility, respectively.

Thanks to Dr. N. Martin and Dr. L. P. Regnault we could perform a Larmor-diffraction experiment; Merci beaucoup!

Dr. N. Qureshi for kindly cutting the samples, which made easier certaindiffraction experiments.

También quisiera agradecer al Prof. J. M. Perez-Mato por todo lo que me haenseñado y por lo que me ha ayudado. El estudio de la supersimetría de las es-tructuras magnéticas incommensurables no habría sido posible sin su dedicación.

Por supuesto, no quiero olvidarme de compañeros y amigos, por las (habit-ualmente largas) discusiones de distinta índole (científicas, políticas, lingüísticas,muchas de ellas totalmente absurdas pero divertidísimas), cafés, cervezas, paseosen las montañas Grenobloises, excursiones . . . en definitiva por haber hecho que es-tos tres años estén llenos de buenos recuerdos: Óscar, Laura C., Laura O., Alberto,Edurne, Carlos, Luiso, Inés, Pedro y otros muchos.

Gracias Jessica por haberme ayudado en mis cortas estancias en Barcelona.

Eskerrik asko Txoli! Nahiz eta urrin egon eta gutxitan ikusi gan, hor zarelakobeti!

Maitetxu! Eskerrik asko hor egotearren. . . eta zelako azken hilabetiak! Ospatubiharra dago! Bixok be nahikua aguantau dou alkarri. Eskerrik asko!!

Mersis Josu. Zure lanaren ondorio be badalako. Lehen aukera zuk emanzoztazulako. Zu barik ezinezkua izango zalako.

Eskerrik bereziena familiari, lan hau zeuon esfortzuaren fruitu be badalako.

Eta orain, uztak batzeko garaia da...

Page 11: Magnetic order, spin-induced ferroelectricity and ... · Magnetic order, spin-induced ferroelectricity and structural studies in multiferroic Mn 1´xCo xWO 4 by IRENE URCELAY OLABARRIA

Preface

The manuscript summarizes the work done during the last three years in the In-stitut Laue Langevin (Grenoble) in collaboration with the Instituto de Ciencia deMateriales de Barcelona, the Universitat Autònoma de Barcelona and the Commis-sariat à l’énergie atomique et aux énergies alternatives (Grenoble).

Since ferroelectricity was found to be coupled to the magnetic order in TbMnO3,there has been a renewed interest in multiferroics, both for fundamental reasonsand for possible technological applications.

About five year ago, it was discovered that the electric polarization thatemerges in MnWO4 is coupled to a cycloidal magnetic structure. This material hasthree magnetic states: AF3 (13.5 K � T � 12.5 K), moments order collinearly along adirection u within the ac plane with a sinusoidally modulated amplitude and an in-commensurate propagation vector k = (-0.214, 1

2 , 0.457); AF2 (12.5 K � T � 6.8 K),it presents an additional magnetic component along b, the propagation vector be-ing the same. Thus, the magnetic ordering is an elliptical cycloidal spin structurewithin the ub plane. In this phase the electric polarization arises along the b axisand therefore, this is the multiferroic phase. AF1 (T � 6.8 K), the magnetic struc-ture is collinear (along u) with k = (˘ 1

4 , 12 , 1

2 ).

The succession of magnetic structures is induced by a strong magnetic com-petition which turns into magnetic frustration. The application of external fieldsor chemical substitution could unbalance the subtle equilibrium between the mag-netic interactions. The work on the thesis was initiated by a publication in the

xi

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literature (on polycrystalline samples) which showed that a subtle substitution ofmanganese by cobalt induces changes and stabilizes the multiferroic structure atlow temperatures. Moreover, new magnetic structures have been determined fromneutron-powder experiments.

The thesis presents a study of the crystal and magnetic structures, and elec-tric polarization of Mn1´xCoxWO4 (x =0, 0.05, 0.10, 0.15 and 0.20) at zero and non-zero magnetic field. We have studied the interplay between the magnetic structureand the electric polarization. To achieve our purposes bulk magnetometry, py-roelectric measurements, neutron- and synchrotron-diffraction experiments havebeen carried out on single crystals.

The manuscript is organized in the following way:

(i) Part I, Introduction: multiferroics, magnetic structures, the studied materialsand the experimental techniques used are introduced. At the end the interestand objectives of the thesis are enumerated.

(ii) Part II, Magnetic, ferroelectric and structural properties of Mn1´xCoxWO4 underzero magnetic field.

(iii) Part III, Magnetic-field-induced transitions.

(iv) Part IV, Summary and Conclusions.

(v) Part V, Appendices.

Throughout the thesis I have participated in all neutron diffraction, syn-chrotron, and many of the bulk magnetic measurements, from the design of theexperiments to the data analysis. The crystals and the electric polarization mea-surements were kindly provided by Prof. A. A. Mukhin, from the Prokhorov Gen-eral Physics Institute (Moscow), with whom we closely collaborated.

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Contents

Index vi

I Introduction 1

1 Multiferroic and Magnetoelectric Materials 3

1.1 Brief historical survey of Multiferroic Magnetoelectric Materials (MMMs) 3

1.2 Characteristics of MMMs . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.3 Types of Magnetoelectric multiferroics . . . . . . . . . . . . . . . . . . 6

1.3.1 Type I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.3.2 Type II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.4 Ferroelectricity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2 Magnetic order 11

2.1 General introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.2 Magnetic anisotropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

i

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2.3 Antiferromagnetism . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.3.1 Antiferromagnetism under field . . . . . . . . . . . . . . . . . 18

2.4 Inverse Dzyaloshinskii-Moriya mechanism . . . . . . . . . . . . . . . 20

2.5 Magnetic frustration . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.6 Description of magnetic structures . . . . . . . . . . . . . . . . . . . . 24

2.6.1 Propagation vector . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.6.2 Representation analysis . . . . . . . . . . . . . . . . . . . . . . 25

2.6.3 Superspace formalism . . . . . . . . . . . . . . . . . . . . . . . 26

3 Mn1´xCoxWO4 29

3.1 MnWO4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.2 CoWO4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3.3 Mn1´xCoxWO4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

4 Experimental techniques 41

4.1 Diffraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

4.1.1 Neutron Diffraction . . . . . . . . . . . . . . . . . . . . . . . . 42

4.1.1.1 Magnetic diffraction . . . . . . . . . . . . . . . . . . . 43

4.1.1.2 Single-crystal diffraction versus powder diffraction . 45

4.1.1.3 Larmor diffraction . . . . . . . . . . . . . . . . . . . . 46

4.1.2 Synchrotron X-ray diffraction . . . . . . . . . . . . . . . . . . . 47

4.1.3 Refinements and agreement factors . . . . . . . . . . . . . . . 48

4.1.4 Instrumentation . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

4.2 Bulk magnetic measurements . . . . . . . . . . . . . . . . . . . . . . . 55

4.3 Pyroelectric measurements . . . . . . . . . . . . . . . . . . . . . . . . . 59

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4.4 Crystal synthesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

5 Interest and Objectives 61

II Magnetic, ferroelectric and structural properties of Mn1´xCoxWO4under zero magnetic field 63

6 Crystal Structure of the Mn1´xCoxWO4 family 65

6.1 The crystal structure of MnWO4 across the magnetic transitions in-vestigated by single-crystal neutron-diffraction . . . . . . . . . . . . . 66

6.1.1 Larmor-diffraction study of the spin-lattice coupling . . . . . 69

6.2 Influence of the Co on the crystal structure of the Mn1´xCoxWO4

family (x = 0.05, 0.10, 0.15 and 0.20) . . . . . . . . . . . . . . . . . . . . 71

6.2.1 Influence of the Co doping on the crystal unit-cell . . . . . . . 73

6.3 Summary and conclusions . . . . . . . . . . . . . . . . . . . . . . . . . 75

7 Detailed description of the magnetic structures of MnWO4 79

7.1 Symmetry analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

7.2 Paraelectric AF3 magnetic phase . . . . . . . . . . . . . . . . . . . . . 86

7.3 AF2 multiferroic phase . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

7.4 Atomic modulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

7.5 Summary and conclusions . . . . . . . . . . . . . . . . . . . . . . . . . 97

8 Magnetic structure determination and ferroelectric properties of Mn1´xCoxWO4

(x = 0.05, 0.10, 0.15 and 0.20) 99

8.1 Mn0.95Co0.05WO4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

8.1.1 Bulk magnetic response . . . . . . . . . . . . . . . . . . . . . . 100

8.1.2 Neutron diffraction . . . . . . . . . . . . . . . . . . . . . . . . . 103

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8.1.3 Electric polarization . . . . . . . . . . . . . . . . . . . . . . . . 105

8.2 Mn0.90Co0.10WO4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

8.2.1 Bulk magnetic response . . . . . . . . . . . . . . . . . . . . . . 106

8.2.2 Neutron diffraction . . . . . . . . . . . . . . . . . . . . . . . . . 108

8.2.3 Electric polarization . . . . . . . . . . . . . . . . . . . . . . . . 111

8.3 Mn0.85Co0.15WO4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

8.3.1 Bulk magnetic response . . . . . . . . . . . . . . . . . . . . . . 112

8.3.2 Neutron diffraction . . . . . . . . . . . . . . . . . . . . . . . . . 115

8.3.3 Electric polarization . . . . . . . . . . . . . . . . . . . . . . . . 120

8.4 Mn0.80Co0.20WO4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

8.4.1 Bulk magnetic response . . . . . . . . . . . . . . . . . . . . . . 122

8.4.2 Neutron diffraction . . . . . . . . . . . . . . . . . . . . . . . . . 123

8.4.3 Electric polarization . . . . . . . . . . . . . . . . . . . . . . . . 126

8.5 x ´ T phase diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

8.6 Evolution of the easy axis . . . . . . . . . . . . . . . . . . . . . . . . . 130

8.7 Evolution of the orientation of the electric polarization . . . . . . . . 132

8.8 Summary and conclusions . . . . . . . . . . . . . . . . . . . . . . . . . 136

9 Lattice anomalies at the ferroelectric and magnetic transitions in cycloidalMn0.95Co0.05WO4 and conical Mn0.80Co0.20WO4 multiferroics 137

9.1 Lattice evolution in Mn0.95Co0.05WO4 . . . . . . . . . . . . . . . . . . 138

9.2 Lattice evolution in Mn0.80Co0.20WO4 . . . . . . . . . . . . . . . . . . 141

9.2.1 Influence of the spiral plane on β at the ferroelectric transition 142

9.3 Summary and conclusions . . . . . . . . . . . . . . . . . . . . . . . . . 144

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III Magnetic-field-induced transitions 147

10 Magnetic-field-induced transition on MnWO4 with field applied along baxis 149

10.1 Summary and conclusions . . . . . . . . . . . . . . . . . . . . . . . . . 158

11 Magnetic-field-induced transitions in Mn0.95Co0.05WO4 159

11.1 Probing field-induced transitions by incremental-magnetic suscep-tibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

11.1.1 Incremental-magnetic susceptibility along intermediate ori-entations between α and α + 90˝ . . . . . . . . . . . . . . . . . 162

11.2 Continuous rotation of the magnetic structure and of the electric po-larization with field aligned parallel to b . . . . . . . . . . . . . . . . . 164

11.2.1 Magnetic order . . . . . . . . . . . . . . . . . . . . . . . . . . . 164

11.2.2 Polarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169

11.3 Magnetic transitions with H ‖ α . . . . . . . . . . . . . . . . . . . . . . 170

11.4 Summary and conclusions . . . . . . . . . . . . . . . . . . . . . . . . . 171

12 Magnetic-field-induced transitions in Mn0.90Co0.10WO4 173

12.1 Bulk magnetic response . . . . . . . . . . . . . . . . . . . . . . . . . . 174

12.2 Neutron diffraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177

12.2.1 Field along the c axis . . . . . . . . . . . . . . . . . . . . . . . . 177

12.2.2 Field along the a axis . . . . . . . . . . . . . . . . . . . . . . . . 179

12.3 Polarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180

12.4 Summary and conclusions . . . . . . . . . . . . . . . . . . . . . . . . . 183

13 Magnetic-field-induced transitions in Mn0.80Co0.20WO4 185

13.1 Magnetic and ferroelectric properties under fields perpendicular to α 185

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13.1.1 Re-orientation of the conical antiferromagnetic structure withH ‖ b . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185

13.1.2 Similar behavior for H ‖ α + 90˝ . . . . . . . . . . . . . . . . . 189

13.2 Stabilization of the cycloidal AF2 phase in H ‖ α configuration . . . . 191

13.3 Summary and conclusions . . . . . . . . . . . . . . . . . . . . . . . . . 194

IV Summary and Conclusions 197

V Appendices 213

A Appendix A: How to perform an experiment on D23 215

B Representation analysis 221

C Constrains of the magnetic modulation due to magnetic superspace sym-metry 225

C.1 P2/c11(α 12 γ)0ss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226

C.2 P2/c11(α 12 γ)00s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229

C.3 P211(α 12 γ)0s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231

C.4 Non-conventional centering X . . . . . . . . . . . . . . . . . . . . . . . 232

D How to intersect the space groups associated to mG1 and mG2 235

E Symmetry of the AF2’ multiferroic phase of Mn0.90Co0.10WO4 239

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Part I

Introduction

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Chapter 1Multiferroic and Magnetoelectric

Materials

The concept of multiferroicity has attracted the attention of a large commu-nity of scientists due to its importance for the solid-state physics as wellas because of possible technological applications that could be developed

based on it. Along the lines of the first chapter, this concept will be introduced,focusing on those multiferroics that exhibit long range magnetic order and ferro-electricity simultaneously.

1.1 Brief historical survey of Multiferroic Magneto-electric Materials (MMMs)

Magnetism and Electricity are both old fields in Physics. First steps in these dis-ciplines were done by ancient civilizations who realized that lodestone and am-ber have amazing properties: both can attract different materials, however ambermust be rubbed with cat’s fur first. They did not understand the origin of thesephenomena, but were able to use those materials for applications such as the com-pass. Nowadays the number of devices based on Magnetism and Electricity iscountless, from very common and simple apparatus to very complex ones. This isa direct consequence of years of study: the more we know and understand the un-derlying mechanisms the more and more complex devices we can build. However,there is still a long way to find out all the mysteries that Magnetism and Electric-

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4 CHAPTER 1. MULTIFERROIC AND MAGNETOELECTRIC MATERIALS

ity hide, even more since it was observed that in some materials the magnetic andelectric properties are coupled, i. e. since magnetoelectrics (and later multiferroics)were discovered.

It is more than a hundred years since the story of magnetoelectric materi-als started. In 1888 Röntgen [1] observed that a dielectric material moving in anelectric field becomes magnetized, and in 1894 Pierre Curie [2], using symmetryconsiderations, foresaw theoretically the possibility of inducing electric polariza-tion in a sample by applying a magnetic field, and vice versa, without having tomove the sample. These are the first two references to magnetoelectric materials.This phenomenon consists of inducing either electric polarization by a magneticfield or magnetization by an electric field. Even if it was known long time ago,there were not experimental proofs until Astrov [3], in 1960, showed that Cr2O3,which is an antiferromagnetic material, is as well magnetoelectric. Astrov’s workwas actually inspired by the paper (published one year earlier) of Dzyaloshinskii[4] predicting, on the ground of symmetry considerations, that magnetoelectriceffect could occur in Cr2O3.

In the 50’s a new and revolutionary concept was born: multiferroicity [5, 6](even though the name was given much later, in 1994, by H. Schmid [7]). By defini-tion, multiferroic materials are those materials that exhibit more than one primaryferroic property simultaneously in the same phase. Ferromagnetism, ferroelectric-ity and ferroelasticity are primary ferroic properties1, and denote materials wheremagnetization, electric polarization or deformation appear spontaneously, are sta-ble and can be reversed following a hysteresis loop if a magnetic, electric or stressfield is applied, respectively.

If a magnetoelectric material is ferromagnetic and ferroelectric in the samephase, it is a Magnetoelectric Multiferroic Material (MMM). One should be awareof the fact that not all the multiferroics are magnetoelectrics and vice versa. Figure 1.1illustrates the classification of the materials taking into account their magnetic andelectric properties.

1.2 Characteristics of MMMs

Intrinsic characteristics of ferroelectricity and ferromagnetism, which are oftenmutually exclusive, make MMM materials rare, in the sense that are not abundant

1Whether ferrotoroidicity is a primary ferroic property or not is still under debate in the community.

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1.2. CHARACTERISTICS OF MMMS 5

Electrically polarizableFerroelectricMagnetically polarizableFerromagnetic

MultiferroicMagnetoelectricMultiferroic

magnetoelectric

FIGURE 1.1: Classification of materials according to the electric and magnetic polarizabil-ity (adapted from Eerenstein et. al. [8]).

in nature [9].

˛ On one hand, each ferroic property has a different symmetry: the electricpolarization is inverted under spatial inversion, magnetization is invertedunder a time-reversal, and ferroelasticity is not inverted under either spatialor time-reversal operations. So, if the crystal structure of a compound con-tains a symmetry operation which reverses, for example, z into ´z, electricpolarization will never show up along the z axis (it would be compensated).In other words, Neumann’s Principle has to be fulfilled: the symmetry elementsof the physical properties of the crystal have to include the symmetry operations ofthe point group of the crystal. Thus, in order to have a multiferroic material it’ssymmetry must allow more than one primary ferroic property.

˛ On the other hand, each property requires different physical conditions thatcould exclude one or the other property. In the case of ferroelectrics, ma-terials are insulators while most of the ferromagnetic materials are metals.Besides that, magnetism usually comes from incomplete d orbitals, whereasferroelectricity often requires d0 orbitals.

Nowadays the definition of multiferroics is more flexible and, frequently, an-tiferromagnetic (which often are insulators) and ferrimagnetic materials are also

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6 CHAPTER 1. MULTIFERROIC AND MAGNETOELECTRIC MATERIALS

considered as potential multiferroics. Therefore, in principle, the primary ferroicproperties are not needed, but long range order. This broadening of the definitionincreases the amount of compounds that are considered as multiferroics and thus,as MMMs.

It is important to stress that unlike classic magnetoelectrics, MMM materialsdo not need any external field to become magnetically and electrically ordered inthe same phase. However, since they are magnetoelectric both order parameterscan be controlled by external fields, either magnetic or electric. This is why theybecome very interesting for technological applications. As an example, one couldthink about electronic devices where the magnetization is controlled by an electricfield. Since electric current is no longer needed, energy losses produced by the heatcoming from the Joule effect are avoided, and overheating of the system as well.For instance, nowadays, reading and writing systems of computers are based onMagnetic Random Access Memory (MRAM) devices but MagnetoElectric RandomAccess Memory (MERAM) are being developed with the hope that one day theycould replace them [10].

1.3 Types of Magnetoelectric multiferroics

Magnetoelectric multiferroic materials are generally classified in two broad groupsaccording to the interplay between ferroelectricity and the magnetic order: type Iand type II [11].

1.3.1 Type I

This type of multiferroics were found first, and up to now they are the most nu-merous [6, 8, 12–22]. Their ferroelectricity and magnetism have different sourcesand appear rather independently. Usually ferroelectricity arises at higher tempera-tures than magnetic order does. This makes them bad candidates for technologicalapplications. However, they are multiferroics at relatively high temperatures (theantiferromagnetic BiFeO3 is a room temperature multiferroic [8]) and their electricpolarization is rather big (10 - 100 μC/cm2), qualities that work in their favor.

Whereas the origin of the magnetic order is always the same (see Chapter 2),there is a large variety of mechanisms that give rise to the ferroelectricity in thesematerials and hence, they can be classified according to that:

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1.3. TYPES OF MAGNETOELECTRIC MULTIFERROICS 7

˛ Perovskite-like structures (ABO3) are very interesting materials where differ-ent mechanisms can be used to induce ferroelectricity. One way is mixing d0

and dn ions in B position and consequently, the resulting ferroelectricity andmagnetic order come from different origins.

˛ Materials with lone-pair atoms (which have outer free electrons), such asBi3+ and Pb2+ are also good candidates for ferroelectrics. Hence, combiningthese ions with magnetic ones may lead to multiferroics. One of the mostfamous multiferroics BiFeO3, belongs to this class [12–14].

˛ Ferroelectricity is often driven by charge ordering (see Ref. [15]), as in per-ovskite manganites [16, 17], magnetite [18–20] and frustrated LuFe2O4 [21]for example. This mechanism is based on the existence of inequivalent sitesand bonds, the origin of which can have diverse nature.

˛ The geometry of the system itself may cause ferroelectricity, as it is the case ofYMnO3 where the rotation of the rigid polyhedra formed by Mn-O producespolarization, see Ref. [22].

1.3.2 Type II

This group is composed by materials in which ferroelectricity is induced by mag-netism. Almost all such materials are multiferroics at relatively low temperatureand show a strong coupling between the magnetism and the ferroelectricity, butthe electric polarization is often weaker than in type I multiferroics.

They were discovered in the last decade and they are at the origin of therenewed interest in this field [23, 29–31]. The strong coupling between the mag-netic ordering and the electric polarization was first reported by Kimura et. al.[23] when they showed that the electric polarization of TbMnO3 rotates when amagnetic field is applied along a specific direction.

One can distinguish at least three subclasses depending on the origin of theelectric polarization: non-collinear multiferroics, exchange-striction multiferroicsand other spin electronic mechanisms.

˛ Non-collinear multiferroics: These are the most numerous type II materials.The ferroelectricity appears related to a specific spiral-like magnetic struc-ture, which usually has a cycloidal component (see Chapter 2). The ferro-

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8 CHAPTER 1. MULTIFERROIC AND MAGNETOELECTRIC MATERIALS

electicity is in this case ruled by the following expression [24–28]:

P „ rij ˆ (Si ˆ Sj) (1.1)

where rij is the vector between two neighboring spins, Si and Sj. Accord-ing to this expression, the electric polarization is always perpendicular tothe spin-chirality vector (Si ˆ Sj). In simple cases it can be translated toP „ k ˆ (Si ˆ Sj), where k is the propagation vector of the magnetic struc-ture. Then, in such cases, only magnetic structures with a cycloidal com-ponent would give rise to electric polarization. However, T. Arima [32] hasshown that electric polarization can also show up in proper-screw structures(k ‖ Si ˆ Sj) in crystals with low symmetry, i. e. triclinic, monoclinic ortrigonal crystals.

Several modulated magnetic structures are depicted in Figs. 1.2(a), 1.2(b) and1.2(c), which are sinusoidal, cycloidal and screw-type structures respectively.In principle, if we restrict to the case where the polarization is linked to thecycloidal component of the magnetic structure, only Fig. 1.2(b) is multifer-roic.

The mechanisms that drive the spin-induced electric polarization are still notfirmly established. There are two main theories that could explain it: theInverse-Dzyaloshinskii-Moriya interaction model (see Section 2.4 inChapter 2) or the spin-current model2. Nevertheless, the role of the magneticfrustration is undeniable, since this is what enforces the complex magneticstructures. All these concepts will be explained in following sections.

˛ Exchange-striction multiferroics: Some examples of collinear magnetic or-ders coupled to ferroelectricity have been found rather recently [33, 38]. Inthese systems the exchange striction is usually the cause of the electric polar-ization. Choi et. al. reported in Ref. [33] that in Ca3CoMnO6 the polar phaseemerges coupled to the magnetic transition into a magnetic structure of thetype ÒÒÓÓ. The different exchange striction between parallel and antiparallelmoments give rise to a ferroelectric distortion. There is still some contro-versy whether RNiO3 nickelates belong to this group or not because thereare several proposed models for the magnetic structure, one of them beinganalogous to the one of Ca3CoMnO6, see [34, 35].

E-type multiferroics consist of zigzag-ferromagnetic chains. Some RMnO3

manganites have this structure. Along the a + c direction the structure is2The spin-current, js, is induced between Si and Sj being both non-parallel which in turn induces

electric polarization. Unlike in the Dzyaloshinskii-Moriya interaction, atomic displacements are notinvolved in this mechanism [27, 28].

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1.4. FERROELECTRICITY 9

ÒÒÓÓ, and the exchange striction causes polar distortions, see Fig. 1.2(e). How-ever, in this scenario the polarization shows up due to transverse displace-ments of oxygen atoms in contrast to previous cases. Moreover, dependingon the strength of the super-exchange or double-exchange interaction the di-rection of the polarization is switched [36, 37].

˛ Other spin electronic mechanisms: Another mechanism that can generate fer-roelectricity is the electronic ferroelectricity observed in frustrated magnets.If S1, S2 and S3 are spins located at the vertices of a regular triangle, thenif S1(S2 + S3) ´ 2S2S3 is nonzero the polarization is nonzero. This situationwas observed by Bulaevskii et. al. in some insulators, see Ref [38].

1.4 Ferroelectricity

Chapter 2 is devoted to magnetic order. Nevertheless, since ferroelectricity is alsoa pillar of this work, we will briefly introduce it this section.

Ferroelectrics consists of an ordered array of electric dipoles being all ori-ented in parallel directions. They are characterized by a spontaneous electric po-larization, P, which is switchable by an applied electric field, E. Ferroelectric P ´ Ehysteresis loops are similar to M ´ H hysteresis loops in ferromagnets. Indeed, thefirst P ´ E hysteresis loop was observed in Rochelle Salt in 1921 [39] and it wasdescribed as equivalent to the magnetic one seen for iron. Thus, the term ferroelec-tricity was used to highlight the analogy. In either cases the macroscopic polar-ization (magnetic/electric) decreases with increasing the temperature and abovethe Curie temperature, TC, the polarization vanishes and the material becomes un-polarized (paramagnetic/paraelectric). When the sample has multiple domains itmay happen that no macroscopic polarization is observed even below TC.

A material to have a must have a polar point-group3 to exhibit spontaneouselectric polarization. Besides that, the electric polarization must be switchable, atransition between two stable states of opposite polarization must be accessible.A fundamental requirement for a material to be ferroelectric is to be an insulator:otherwise an applied electric field would induce a current rather than reorient itspolarization.

3Very often instead of the term polar it is used non-centrosymmetric point group which is a necessarycondition but not sufficient. Note that for example 222 point group is neither centrosymmetric norpolar.

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10 CHAPTER 1. MULTIFERROIC AND MAGNETOELECTRIC MATERIALS

P ≠ 0

Si x Sj

k(a)

(b)

(c)

(d)

(e)

k || Si x Sj

k

a

c

P = 0

P = 0

P ≠ 0Co+2 Mn+4

P ≠ 0

FIGURE 1.2: Sketch of magnetic structures. (a) Non-polar sinusoidally modulatedcollinear structure. (b) Cycloidal structure with electric polarization in the plane wherethe moments rotate. (c) Screw-type structure, which in principle is non-polar. (d) Situ-ation where the exchange striction gives rise to electric polarization in a collinear struc-ture. Dotted circles represent the original position of Co2+ before the transition to theferroelectric states that occurs in Ca3CoMnO6. (e) Projection along the b axis of a RMnO3

manganite where the E-type magnetic structure is visible.

Antiferroelectrics consists of an ordered array of electric dipoles, but withadjacent dipoles oriented in opposite directions [41, 42]. In an antiferroelectric, themacroscopic polarization is zero, since the adjacent dipoles cancel each other out.They are analogous to antiferromagnets.

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Chapter 2Magnetic order

Along the lines of this chapter the reader will find a brief introduction tothe magnetic order: origin of the magnetic order, types of structures andthe different methods to describe them.

2.1 General introduction

Magnetism is a direct consequence of the quantum nature of materials: spins andPauli’s exclusion principle are on its base. The magnetic ordered state is the eigen-vector of the following Hamiltonian:

H = HCoulomb +HS´O +HCristalField (2.1)

where HCoulomb takes into account the Coulomb interaction between non-coupledelectrons, HS´O = (Λ/h/2)L ¨ S refers to the spin-orbit interaction and HCristalField

is the so-called crystal-field interaction which introduces the effect of the crys-talline structure in the magnetic order, i. e. the Coulomb interaction of the elec-tronic charge distribution ρ0(r) of an ion embedded in a solid with the surroundingcharges in the crystal.

The total Hamiltonian, H, can be simplified, within the Heisenberg model,as a function of the spins as:

H = ´ÿij

JijSi ¨ Sj (2.2)

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12 CHAPTER 2. MAGNETIC ORDER

where Jij refers to the exchange constant between Si and Sj (explicitly located atthe atom positions) and is a measure of the strength of the exchange interactions:Jij ą 0 indicates a ferromagnetic interaction, which tends to align the two spinsparallel; Jij ą 0 indicates an antiferromagnetic interaction, which tends to alignthe two spins antiparallel. Below certain temperature, those magnetic interactionscould eventually "freeze" the spins so that the materials become magnetically or-dered.

As mentioned in Chapter 1, one necessary condition for magnetic order is tohave unpaired electrons (e.g. 3d, 4 f elements). However at high enough tempera-ture the order is destroyed by the thermal fluctuations. In this paramagnetic statethe response of the randomly fluctuating magnetic moments to an external field,i. e. the magnetic susceptibility χ, usually follows the Curie-Weiss law:

χ =C

T ´ ΘP(2.3)

where C stands for the Curie constant (9μ2Bm2

e f f ) and ΘP for the paramagnetictemperature. The latter is directly related to the exchange interactions.

Below, a brief oversimplified picture of the main types of magnetic structuresand their principle characteristics is given. Figure 2.1 illustrates those structuresand how their magnetization/susceptibility change with temperature.

Ferromagnets: The exchange interaction between neighbor spins is positive, Jij ą 0,and consequently, the magnetic moments have parallel alignment below theCurie temperature, TC. Therefore, in the ordered state there is spontaneousmagnetization, even in absence of any external magnetic field. The mag-netic susceptibility tends to infinity (following the Curie-Weiss law) at theparamagnetic temperature, Θp, that ideally is equal to TC, the point at whichspontaneous magnetization emerges [equation (2.3)].

(a)

Tc ≈ 𝛩

𝜒-1M

TN

𝜒-1

𝛩P𝛩P Tc

𝜒-1M(b) (c)

FIGURE 2.1: Temperature dependence of magnetization, M, and the inverse of magneticsusceptibility, χ, for (a) ferromagnets, (b) ferrimagnets and (c) antiferromagnets.

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2.1. GENERAL INTRODUCTION 13

Ferrimagnets: The exchange interaction is negative, Jij ă 0, rending antiparal-lel alignment of the moments, but the neighboring atoms do not have equalspin, and hence do not compensate, in other words, these materials exhibitantiparallel alignment of non-equivalent neighbor moments below TC. Onecan consider that there are two different ferromagnetic sublattices that inter-penetrate each other in an antiparallel way and thus, the magnetic momentsdo not cancel resulting in net magnetization. In this circumstances, the sus-ceptibility tends to infinity at the Curie temperature as well, but the para-magnetic temperature ΘP, which is extrapolated from its behavior at hightemperatures according to equation (2.3), does not coincide with the Curietemperature. Approaching TC, the inverse susceptibility decreases in a con-cave fashion.

Antiferromagnets: If the sublattices in a ferrimagnet have equal moments, the ma-terial is an antiferromagnet. Thereby, the total magnetization is canceled outand no spontaneous magnetization arises in the material when it becomes or-dered below the Néel temperature, TN . The paramagnetic temperature, Θp,is usually negative and fulfill TN = |ΘP| which is a straight forward conse-quence of considering just first neighbors antiferromagnetic interaction. Thisis discussed in more details in Section 2.3. The magnetic susceptibility has amaximum at TN (� ΘP).

Apart from the three situations introduced above, which can be consideredas simple structures, there are many others such as canted ones, amplitude mod-ulated structures, circular helixes, elliptical helixes, cycloidal structures, conical,transverse conical structures . . . (some examples are depicted in Fig. 2.2). Notethat the difference between a helix and a cycloid lies in the relation between therotation plane of the spins and the direction of the propagation vector k: in a cy-cloidal structure k is within the rotation plane, whereas in helical structures theyare perpendicular to each other. Although most of these exotic structures are veryoften considered as antiferromagnets (because they do not show any spontaneousmagnetization and the magnetic susceptibility has a similar temperature depen-dence) it is clear that the previous classification is not precise enough for all thescenarios that arise in reality. Many of them are met with rare-earth elements.

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14 CHAPTER 2. MAGNETIC ORDER

(a)

(b)

(c)

(d)

FIGURE 2.2: (a) Helical, (b) conical, (c) cycloidal and (d) sinusoidally modulated magneticstructures.

2.2 Magnetic anisotropy

Even if the exchange interactions are isotropic, spins tend to position along cer-tain directions, i. e. along the easy axes or easy directions, which may or may notcorrespond to crystallographic directions. Therefore, the isotropic nature of theexchange interaction is shadowed by the effect of intrinsic and extrinsic proper-ties of the crystals or samples that break such isotropy and fix certain directionsenergetically more favorable.

There are several kinds of anisotropies: magnetocrystalline anisotropy, shapeanisotropy (note that there is no shape anisotropy in an antiferromagnet, becausethere is no demagnetizing field when M = 0), stress anisotropy, induced anisotropy(by magnetic annealing, plastic deformation or irradiation) and exchange anisotropy.Among them, magnetocrystalline anisotropy is the unique one which is intrinsic tothe material, and reflects the symmetry of the crystal. It is mainly driven by thespin-orbit coupling in systems where the atomic orbitals and crystal lattice arestrongly coupled. The orientation of atomic orbits are fixed to the lattice and theenergy barrier to change their orientations entails the application of large fields.Therefore, by means of spin-orbit coupling the moments become linked to the lat-tice. Depending on the strength of the latter coupling, the crystal anisotropy ismore or less important. In other words, the HCristalField term of equation (2.1) is

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2.2. MAGNETIC ANISOTROPY 15

responsible for the magnetocrystalline anisotropy, and it is expressed as follows

HCristalField =

żρ0(r)φc f (r)d3r (2.4)

where φc f is the potential produced by the distribution of charge ρ0(r1) in the restof the crystal at the atomic position r. This crystal field interaction tends to stabilizeparticular orbitals and the spin orbit coupling ΛL ¨ S aligns the spins dependingon the orientation of the orbitals.

Lets consider the case of cobalt, which has hexagonal symmetry. The easy di-rection of magnetization is the c crystallographic direction of the hexagonal close-packed structure, being all directions in the basal plane equally hard. Therefore,the magnetic anisotropy is uniaxial and the anisotropy energy, Ean, when trying totake out the moments from this direction can be expressed as a function of a singleangle, θ, the angle that moments make with respect to c. The anisotropy-energyfunction must be even, because the energy should not depend on the direction ofthe moment within a certain axis and it has to be symmetric with respect to theangle. Hence, the energy can be expressed as follows:

Ean = K0 + K1 sin2 θ + K2 sin4 θ + . . . (2.5)

where K0, K1 and K2 are constant values particular for each material and are tem-perature dependent. Equation (2.5) is general for any system with uniaxial anisotropyand the relation between K1 and K2 fixes the orientation of the easy axis. Higherorder terms may be needed for certain materials, where the dependence with thespherical angle φ appears.

In the simplest of the lowest order case the anisotropy energy can be consid-ered uniaxial and can be expressed as follows:

Ean = Ku sin2 θ. (2.6)

The field needed to saturate the magnetization along the hard direction, theso-called anisotropy field (Han), depends on the strength of the anisotropy and inthe simplest cases it can be expressed as

Han =2Ku

μ0M. (2.7)

As a result of the magnetic anisotropy the magnetic properties will gener-ally depend on the direction in which they are measured. Broadly speaking whenmagnetic field is applied along the easy axis the magnetization is saturated at low

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16 CHAPTER 2. MAGNETIC ORDER

fields (high enough to erase possible magnetic domains which appear as a result ofminimizing the magnetoestatic energy in sample with finite dimensions), whereasalong the hard axis higher fields are requested to align the moments along thefield. Magnetic anisotropy is also responsible for the type of magnetic transitionsthat undergo the materials when an external field is applied along different mag-netic axis. All this will be further discussed in Section 2.3.

The effect of the magnetic anisotropy is not only property of the magneticallyordered phase, but also of the paramagnetic state [43, 44]. In the case of free ions,as for example in a paramagnetic gas, with an angular momentum j all the 2j + 1states will have the same energy in the absence of any field. This degeneracy, maybe removed by a magnetic field, which splits the single energy level into 2j + 1nearly equidistant levels by the so-called Zeeman effect. The resulting ion distri-bution gives a subsequent magnetic moment along the field (see Section 2.3.1). Onthe contrary, if the ions are not initially free, as for example in a crystal, and aresubject of an electric field (the crystal field), the degeneracy in respect of the orbitalmoment, l, may be split by Stark effect1 and therefore, the l moment tends to havea defined orientation in the field which interferes its response to an external mag-netic field. The nature of the splitting will depend on the initial states involved,the nature, distances and symmetry of the surrounding atoms. The symmetry ofthe crystalline field surrounding a paramagnetic ion is not necessarily similar tothe one of the crystal as a whole. Unless the crystalline field possesses cubic sym-metry, one may expect not only magnetic anisotropy but also different values ofthe paramagnetic temperature θ [45].

2.3 Antiferromagnetism

As introduced in Section 2.1, negative exchange interaction Jij ă 0 gives rise ei-ther to antiferromagnets or ferrimagnets. From now on the matter will be focusedon antiferromagnets. In 1936 Louis Néel suggested the scenario of two equal sub-lattices aligned in an antiparallel manner to explain a magnetic transition signedby a small peak in the magnetic susceptibility and an anomaly in the specific heatmeasurements, without showing any net magnetization below the Néel tempera-ture, TN [46]. In the 50s this was proved by neutron-diffraction experiments whichdirectly measured the magnetization of each sub-lattice in MnO [47].

1It is analogous to the Zeeman effect but the splitting of a spectral line into several componentsoccurs in the presence of an electric field.

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2.3. ANTIFERROMAGNETISM 17

The molecular field theory which explained the behavior of ferromagnetswas extended to antiferromagnets. Within this theory, if each magnetic sub-lattice,A and B, have MA = ´MB magnetization, nAB and nAA represent the intersublat-tice and intrasublattice molecular field couplings, respectively, and H0 is an exter-nal magnetic field, then the magnetic field felt by each sub-lattice is

HA = nAAMA + nABMB + H0,

HB = nBAMA + nBBMB + H0. (2.8)

where nAA = nBB and nAB = nBA. In the magnetically ordered state, T ă TN ,at zero external field (H0 = 0) the net magnetization is zero, however the spon-taneous magnetization of each sub-lattice is governed by the Brillouin function asfollows:

MA,B = ˘MA0n2

gμB JB(xA,B) (2.9)

where xA,B =μ0gμB J|HA,B|

kBT and n is the number of magnetic ions per volume unit.

In the paramagnetic state, T ą TN , the magnetization is zero unless an exter-nal field is applied. In such situation, the induced magnetization behaves accord-ing to

MA,B = χHA,B, (2.10)

where χ = C1T =

nμ0m2e f f

6kBT . At H0 = 0 and when the system reaches TN , the spon-taneous magnetization emerges, consequently, the relation between nij coefficientsmust be such that

(C1TN

nAA ´ 1)2 ´ (C1TN

nAB)2 = 0 (2.11)

which derives in

TN = C1(nAA ´ nAB). (2.12)

Thereby, the Néel temperature is expressed as a function of the intrasublattice andintersublattice interactions. In the presence of external magnetic field, the totalmagnetization of an antiferromagnetic material in the paramagnetic state mustagree with the sum of the magnetization of both sub-lattices:

M =C

T ´ ΘPH0, (2.13)

being C = 2C1 and ΘP = C1(nAA + nAB) the paramagnetic Curie temperature,within the simple model were only intersublattice nAA and intrasublattice nAB

couplings. When, only interactions between nearest neighbors are involved in themagnetism of the antiferromagnetic material, nAA, ΘP = ´TN .

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18 CHAPTER 2. MAGNETIC ORDER

2.3.1 Antiferromagnetism under field

When a material is under magnetic field, another term has to be introduced tothe magnetic Hamiltonian of the system, the Zeeman term, and equation (2.1) be-comes:

H = HCoulomb +HS´O +HCristalField +HZeeman (2.14)

where the Hamiltonian term for the Zeeman effect is written as:

HZeeman = (μ0μB/h)(L + 2S) ¨ B (2.15)

The antiferromagnetic axis along which the two sub-lattice magnetizationslie is determined by magnetocrystalline anisotropy, and the magnetic response be-low TN depends on the direction of H relative to this axis. If a small field is appliedalong its easy axis, a net induced magnetization will be twice as big as the differ-ence of the magnetization in each sub-lattice δM,

M = M1A + M1

B = 2δM (2.16)

and the magnetic susceptibility, along the easy axis can be derived assuming thatthe intrasublattice coupling is negligible, nAA = 0, and using the Taylor series ofthe Brillouin function around H0 = 0:

χ‖ =M1

A + M1B

H0=

2δMH0

=2C1[3J/(J + 1)]BJ 1(x0)

T ´ nABC1[3J/(J + 1)]BJ 1(x0). (2.17)

When T Ñ 0 the susceptibility along the easy axis tends to zero,

χ‖(T Ñ 0) = 0, (2.18)

and at TN , where MA,B = 0 and BJ 1(0) = J+13J ,

χ‖(T = TN) = ´ 1nAB

. (2.19)

If the same field is applied along the hard axis, then the magnetic momentsof both sub-lattices tilt towards the external field making a δ angle with the easyaxis, and the total magnetization is M = 2MA sin δ. The moments of the sub-lattices will rotate until the molecular field equals the external field, and thereforeH = 2nAB MA sin δ. Then,

χK = ´ 1nAB

. (2.20)

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2.3. ANTIFERROMAGNETISM 19

In this oversimplified model, the susceptibility along the hard axis does notdepend on the temperature below the Néel temperature. In Fig. 2.3(a) are plottedχ‖ and χK of a simple antiferromagnet.

These results are obtained in the very simple case where only interactions be-tween nearest neighbors are considered (nAA = 0 and nAB � 0) and the anisotropyconstants, Ku in uniaxial systems, is considered to be nAB ą Ku

2M2 . However, of-ten the behavior of the susceptibility in antiferromagnets is more complicated, andequations (2.18), (2.19) and (2.20) are no longer applicable. In the case of modu-lated structures, it has been seen that χ‖(T Ñ 0) ą 0 and χK(T ă TN) ă ´ 1

nAB, see

2.3(b). This is the case of screw-type2 structures [48–50]. In such cases and under asmall external field parallel to the rotation plane, the magnetic susceptibility χscrew

‖has a finite value at 0 K [see Fig. 2.3(b)]. If, on the contrary, the field is applied per-pendicular to the rotation plane of the moments, the magnetic susceptibility χscrewKat 0 K is smaller to that of conventional collinear antiferromagnets, as depicted inFig 2.3(b). Nevertheless, both χscrew

‖ and χscrewK have the same value at TN .

T T

𝛘 𝛘

(a) (b)𝛘⊥

𝛘||

𝛘⊥screw

𝛘||screw

TN TN

|1/nAB|

FIGURE 2.3: Magnetic susceptibility on an antiferromagnet along the easy and hard axes(a) within the molecular field theory and considering that nA A and Ku are negligible and(b) in more complicated systems such as magnetically modulated structures.

Magnetic structures can be modified/changed when high enough fields areapplied to the magnetically ordered materials and hence, magnetic transitions canbe induced. In principle, according to the Zeeman term [see equation (2.15)] themagnetization should be always perpendicular to the external field, but this willdepend on the strength of the anisotropy field [equation (2.7)]. In the simplest case,when the field is applied along the easy axis, the moments will turn perpendicularto the field when the energies of the parallel and perpendicular configurations are

2Proper screws are the so-called helixes (k Krotation plane) and improper screw structures are theso-called cycloidal structures (k ‖rotation plane)

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20 CHAPTER 2. MAGNETIC ORDER

the same, e. i. when

´ μ0MHan ´ μ0χ‖H2s f = ´μ0χKH2

s f , (2.21)

Hs f =

dMHan

χK ´ χ‖. (2.22)

When the spin-flop field, Hs f , is reached both sub-lattices are perpendicularto the field, they will rotated towards the field. This process is known as a spin-floptransition, and it is sketched in Figs 2.4(a), 2.4(b) and 2.4(c). Magnetization curvesalong easy axis is shown in Fig. 2.4(d). In spin-flop-like transitions, the magneti-zation (M‖) increases very slowly, according to χ‖, until a threshold field when themoment become perpendicular to the field and then, it increases, according to χK,until it saturates (moments parallel to the field). This saturated high field phase,in the western literature is termed "paramagnetic" while in the Russian literature aterm "forced ferromagnetism" is used.

The situation is slightly different when the anisotropy is very strong. At lowfields, magnetization increases with H according to the value of χ‖. Then, at acritical value Hs f lip, the antiparallel spins flip over and become parallel, in otherwords, the situation jumps directly from Fig. 2.4(a) to Fig. 2.4(c), and the magne-tization saturates. Since the magnetic anisotropy is very strong, the intermediateconfiguration Fig. 2.4 (b) is not stable. The spin-flip and spin-flop transitions areof metamagnetic type ones. Usually the critical field is rather high. The magne-tization curves along easy and hard axis in this kind of transitions are shown inFig. 2.4(e).

2.4 Inverse Dzyaloshinskii-Moriya mechanism

The Dzyaloshinskii-Moriya (DM) interaction is an antisymmetric, anisotropic ex-change coupling between two spins on a lattice bond ij with no inversion center[24, 25]. For spins Si and Sj, a new term in the Hamiltonian is then given by

HDM = Dij ¨ (Si ˆ Sj) (2.23)

where the Dzyaloshinskii-Moriya vector, Dij, is finite when the crystal field doesnot have inversion symmetry with respect to the center between Si and Sj. Theeffect of the DM interaction is often to provide a small canting of the momentsin an antiferromagnetic structure, resulting in weak ferromagnetism. The DM in-teraction favors non-collinear spin ordering, as typically in the weak ferromagnet

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2.4. INVERSE DZYALOSHINSKII-MORIYA MECHANISM 21

H HH

M M

(a) (b) (c)

(d) (e)

M⊥

M||

M⊥

M||

H

FIGURE 2.4: (a) Antiferromagnetic spin-configuration. (b) Spin-flop configuration.(c) Forced ferromagnetic (also called paramagnetic) configuration that describes the fi-nal stage in metamagnetic transitions. Magnetization curves with field parallel to the easyaxis (M‖) in (d) spin-flop and (e) spin-flip transitions. Magnetization curves with fieldperpendicular to the easy axis (MK) are also shown.

R2CuO4 cuprates [52]. Generally, in DM weak ferromagnets the non-collinear or-der is not at the origin, but it is a consequence of the structural distortion.

𝟇rij

Si Sj

Mn1 Mn2

O

x

P

FIGURE 2.5: Electric polarization induced by the the displacement of the oxygen atomsdue to the DM interaction.

Sergienko et. al. proposed a relationship between the DM interaction andthe onset of ferroelectricity in systems, such us TbMnO3 [26], where the electric

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22 CHAPTER 2. MAGNETIC ORDER

polarization appears together with a non-collinear spin order. That is the so-calledInverse Dzyaloshinskii-Moriya (IDM) mechanism. According to it, the magneticcoupling energy triggers a structural distortion resulting in a polar structure. Inthe IDM mechanism, Dij is non-zero in the magnetically ordered phase, in con-trast to the DM mechanism (Dij can be non-zero above the transition to the weakferromagnet phase). The vector Dij shown in equation (2.23) is proportional to(x ˆ rij) and the spin-orbit coupling, where x is the displacement of the oxygenatom (see Fig. 2.5). The corresponding energy is therefore increased by pushingthe oxygen anions away from the positively charged Mn ions, resulting in anelectric polarization. As the displacement x of the oxygen atoms increases, theMn-O-Mn bond angle φ decreases, suggesting an important relationship betweenφ and the existence of multiferroic properties. Goto et. al. suggested a relationshipbetween the multiferroic properties seen in TbMnO3 and DyMnO3 and φ [51]. Itwas apparent that the cycloidal magnetic order was only seen for φ in the range144.5˝ � φ � 145.8˝, implying a strong relationship between the relative positionsof neighboring Mn atoms and the magnetic frustration they experience.

2.5 Magnetic frustration

In frustrated magnets the magnetic interactions can not be satisfied simultane-ously. In other words, the competing interactions in the Hamiltonian contributein such way that the energy can not be minimized in a single ground state [53] andit becomes degenerated.

The geometry of the crystal or the competing interactions between differentSi and Sj couples may lead to complex structures.

To illustrate the case where a variety of Ji play an important role in the expres-sion (2.2), lets imagine a square lattice where the interaction between the nearestneighbors is J1 and between the next nearest neighbors is J2, both being antifer-romagnetic interactions. Depending on the relation between the strength of thesetwo interactions either of the two spin configuration depicted in Figs. 2.6(a) and2.6(b) will be stabilized. Thus, in the situation where the |J1| ą |2J2|, first neighborswill be antiferromagnetically coupled whereas second neighbors will be forced tobe parallel [Fig. 2.6 (a)]. On the other hand, if |J1| ă |2J2| the scenario would be theopposite, where first neighbors will be forced to be parallel and second neighborsantiparallel. There is a third possibility to take into account, and for us the mostinteresting: the situation where the interactions are comparable, 2J2 „ J1. In that

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2.5. MAGNETIC FRUSTRATION 23

framework the structure becomes magnetically frustrated.

(c) FM coupling

(d) AFM coupling

?

(a) J1>2J2 (b) J1<2J2

J2

J1

J2

J1

FIGURE 2.6: (a) and (b) show possible spin configurations depending on the relationshipbetween first and second neighbors antiferromagnetic exchange interactions in a squarelattice. If the interaction are comparable, 2J2 „ J1, then the system would be frustratedand the ground state degenerated. In (d) the frustration emerges from the geometry ofthe spin lattice even considering just first neighbors. Whereas a ferromagnetic coupling(c) is compatible with the lattice; (d) the antiferromagnetic it is not and this gives rise todifferent states. Adapted from reference [53].

In other cases, the origin of the frustration may be in the topology of the sys-tem, being all the exchange interactions equal. For instance, one can think abouta triangular spin lattice, as shown in the Figs. 2.6(c) and 2.6(d). If the interactionbetween nearest neighbors are ferromagnetic, they can all be satisfied simultane-ously [Fig. 2.6(c)] and the ground state is unique. However, if the interactionsbetween the nearest neighbors is antiferromagnetic, as in Fig. 2.6(d), the systemdoes not have a unique spin configuration that minimizes the energy and, hence,the ground state becomes degenerated and several spin structures may arise in thesystem.

There are distinct types of lattices and atomic configurations that favor themagnetic frustration, such as triangular lattices, the Kagomé lattice, pyrochlorelattice . . .

In these systems TN = |ΘP| is no longer true because the interaction thatplay a role a not only those which concern between nearest neighbors. It has been

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24 CHAPTER 2. MAGNETIC ORDER

suggested that the |ΘP|TN

ratio could be used to characterize the degree of frustra-tion, the more frustrated the system is the larger the value of the ratio is [55]. Itdoes not mean that all the systems that show TN ă |ΘP| are frustrated, as forexample Mn [56].

2.6 Description of magnetic structures

As we have previously seen, very often it is not sufficient to say that a certain mag-netic structure is either ferromagnetic, antiferromagnetic or ferrimagnetic, thus acomplete and detailed description of the structure is mandatory.

In general, being the magnetic structure a periodic function, the orientationof the magnetic moment of an atom j in the unit cell l can be Fourier expanded anddescribed by the following sum:

ml j =ÿk

Skje´2πik¨Rl (2.24)

where Skj stands the Fourier components of the distribution associated to thepropagation vector k.

As in conventional crystallography, magnetic structures are characterized bytwo features: the periodicity of the structure and the motif.

One the one hand, the first difficulty when dealing with magnetic structuresarises from the fact that even if they are periodic, these structures do not necessar-ily have the same periodicity as the underlying crystallographic structures. In thethree main cases seen in the previous section, ferromagnetic, antiferromagnetic orferrimagnetic, the periodicity of the magnetic structure is equal or is an integralmultiple of the nuclear one. However, this is not always the case and therefore, amore rigorous and better adapted method is used to describe magnetic structures:the propagation vector formalism, k, see Section 2.6.1.

On the other hand, the magnetic motifs within the magnetic cell (propagatedthroughout the crystal according to k) could correspond to any arrangement ofthe moments. However, as mentioned in the first chapter, the arrangement of thespins have to contain the symmetry of the crystal. Therefore, symmetry consid-erations can help to reduce the number of possible magnetic structures. Thesesymmetry considerations can be done mainly through two formalisms: representa-tion analysis (see Section 2.6.2) and superspace formalism (see Section 2.6.3). These

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2.6. DESCRIPTION OF MAGNETIC STRUCTURES 25

two formalisms impose sets of constraints that are closely related, but superspacesymmetry is in general more restrictive and more comprehensive, as it affects allthe degrees of freedom of the system.

2.6.1 Propagation vector

The propagation vector, k, is a mathematical object defined in the reciprocal spacewhich describes the relation between the orientation of the equivalent atoms in dif-ferent nuclear cells. Hence, from equation (2.24) it follows that k = (0, 0, 0) whichmeans that the magnetic moments of equivalent atoms in two contiguous nuclearcell are parallel and consequently the magnetic cell coincides with the nuclear one.Let us assume that the situation is such that along a the moments are antiparallel,then k = (1

2 , 0, 0) and the magnetic unit cell would be doubled along the a axiswith respect to the nuclear, i. e. amag = 2anucl . In other words, k = (ka, kb, kc) rep-resent the wave vector that defines the phase difference between the moments of amagnetic modulation. When (ka, kb, kc) are rational numbers, then the structure iscommensurable, and if not, it is incommensurable with period 1.

Magnetic structures can be described not only by one propagation vector(single-k structures), but by various (multi-k structures). The most illustrative caseof a multi-k structure is a canted one as represented in Fig. 2.2. This is a cantedstructure where it is mandatory to use two propagation vectors to completely de-scribe the relation between magnetic moments in different cells: the antiferromag-netic coupling is described by k = (1

2 , 0, 0)-like vector whereas the ferromagneticone needs k = (0, 0, 0).

2.6.2 Representation analysis

This method is widely used. Once, the propagation vector is known, symmetryconsiderations and group theory can reduce the number of possible solutions. Themagnetic modulations can be described by the representation analysis method (de-veloped by Bertaut [57, 58]) which is based on the decomposition of the magneticconfiguration space into basis modes transforming according to different physi-cally irreducible representations, the so-called irreps, of the paramagnetic spacegroup (regardless of their propagation vector being commensurate or incommen-surate).

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26 CHAPTER 2. MAGNETIC ORDER

In the phase transitions that occur due to a break of the symmetry of the sys-tem and are characterized by an order parameter η, above the transition, the orderparameter is zero and below a critical temperature, T0, it can vary continuously ordiscontinuously (second and first order transitions, respectively) [59]. In secondorder transitions, in the paramagnetic state η = 0 and at T0 it starts increasingsmoothly. The number and types of order parameters can be derived from grouptheory since they are second-order invariants and group theory tells that second-order invariants are formed by components of basis vectors belonging to the sameirreducible representation Γv of the paramagnetic group Gp, which leaves the sys-tem invariant above the transition. Thus, if the magnetic transition is second order,only one irreducible representation Γv of Gp becomes critical at T = T0.

The complete representation analysis of MnWO4 is given in Appendix B.

2.6.3 Superspace formalism

In 1980 A. Janner and T. Janssen proposed the use of superspace formalism todescribe incommensurate magnetic structures [60]. However, magnetic structureshave not been studied in the light of this formalism until the program JANA2006 [61],which allows to perform refinements using symmetry constraints consistent withany magnetic superspace group [62, 63].

The tensor properties of a crystal are constrained by the magnetic point group.Superspace formalism provides, in a simple manner, not only the symmetry of themagnetic structures and their intrinsic restrictions, but also information about thetensor properties of each incommensurate phase, such as ferroelectricity or magne-tostructural properties [64–67]. This fact makes the formalism very appropriate tostudy multiferroic properties, since it allows to rationalize the physical propertiesinduced by the incommensurate magnetic order in the framework of a symmetrybreak.

Within the superspace formalism, a modulated magnetic structure with asingle incommensurate propagation vector k is described by a periodic structureplus a series of atomic modulation functions that define the deviations of eachatom from the periodic structure3. In general, a modulation function with period1, Aμ(x4), describes the variation of the property Aμ of an atom μ from one cell ofthe periodic structure to another, so that its value in the unit cell l is written the

3The basic concepts of this formalism are summarized from the review paper of Perez-Mato et. al. [66].

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2.6. DESCRIPTION OF MAGNETIC STRUCTURES 27

following manner:Alμ = Aμ(x4 = k ¨ rlμ) (2.25)

with the atom located in the basic position rlμ = l + rμ (l being a lattice translationof the basic structure).

These atomic modulation functions, with period 1, can be expanded in aFourier series of the type:

Aμ(x4) = Aμ,0 +ÿ

n=1,...

[Aμ,ns sin(2πnx4) + Aμ,nc cos(2πnx4)] (2.26)

By definition, if (R, θ |t) is an operation of the magnetic space group Ωb ofthe basic structure 4 it transforms the incommensurately modulated structure intoa distinguishable incommensurate modulated structure: the same basic structureplus all its modulation functions changed by a common translation of the internalcoordinate x4. In other words,

(R, θ|t)Aμ(x4) = A1μ(x4) = Aμ(x4 + τ). (2.27)

The translation τ depends on each specific operation. This implies that theoriginal modulated structure can be recovered by performing an additional trans-lation τ along the so-called internal coordinate, i.e. the phase of the modulationfunctions. In this sense, one can speak of (R, θ |t, τ) as a symmetry operation ofthe system defined in a four-dimensional mathematical space.

If (R, θ |t, τ) belongs to the superspace group of an incommensurate mag-netic structure with a single propagation vector, the action of R on its propagationvector k transforms this vector into a vector equivalent to either k or ´k.

k ¨ R = RIk + HR, (2.28)

where RI is either +1 or -1 and HR is a reciprocal lattice vector of the basic structurethat depends on the operation R. And the atomic modulation functions of thesymmetry-related ν and μ atoms ((R |t)rν = rμ + l) are restricted as follows:

Aμ(RI x4 + τ0 + HR ¨ rν) = Trans f (R, θ)Aν(x4), (2.29)

where τ0 = τ + k ¨ t and Trans f (R, θ) is the operator associated with the trans-formation of the local quantity Aμ under the action of the point-group operation

4with R being a point-group operation, θ being -1 or +1 depending if the operation includes timereversal or not, and t a translation in 3d real space

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28 CHAPTER 2. MAGNETIC ORDER

(R, θ). Thus, equation (2.29) introduces a relationship between the modulationfunctions of the magnetic moments of the two atoms:

Mμ(RI x4 + τ0 + HR ¨ rν) = θdet(R)R ¨ Mν(x4). (2.30)

The four elementary translations (1,+1|100, ´kx), (1,+1|010, ´ky),(1,+1|001, ´kz) and (1,+1|000, 1) generate the (3+1)-dim lattice and define a unitcell in the (3+1)-dim superspace. In this basis the symmetry operation (R, θ |t, τ)

can be expressed in the standard form of a space group operation in a 4-dim space,(Rs, θ |ts), where ts is a 4-dim translation and Rs a 4 ˆ 4 integer matrix definingthe transformation of a generic point x1, x2, x3, x4:

Rs =

⎛⎜⎜⎜⎜⎝

R11 R12 R13 0R21 R22 R23 0R31 R32 R33 0HR1 HR2 HR3 RI

⎞⎟⎟⎟⎟⎠ , (2.31)

where Rij are the matrix coefficients of the rotational 3-dim operation R belongingto Ωb (expressed in the basis of the basic unit cell) and (HR1, HR2, HR3) are the com-ponents of the vector HR defined in equation (2.28). The superspace translation ts

in the 4-dim basis is given by (t1, t2, t3, τ0), where ti are the three components of tin the basis of the basic unit cell.

Summarizing, an incommensurate magnetic structure can be fully describedby specifying:

(i) its magnetic superspace group.

(ii) its periodic basic structure.

(iii) and a set of periodic atomic modulation functions that define the magneticmodulations for the atoms of the asymmetric unit of the basic structure.

Note that the magnetic point group of the system is given by the set of all point-group operations present in the operations of this superpace group.

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Chapter 3Mn1´xCoxWO4

Zure iparrak noraezaren lekukoa hartuzuen

Mikel Urdangarin (1971 -)

T he compounds studied in this thesis will be introduced to the reader inthe following chapter. First, the properties of both MnWO4 and CoWO4,are described. Then, the reasons for studying the Mn1´xCoxWO4 solid

solution are summarized.

3.1 MnWO4

Manganese Hübnerite, MnWO4, was discovered in 1852 by A. Brelthaupt whonamed it as Megabslte. In 1865 it was renamed as Hubnerite in honor of the metal-lurgist Adolf Huebner. It is a dark, reddish, transparent mineral with one directionof easy cleavage [69]. It is a semiconducting material [70, 71], that has photolu-minescence properties [72] and can be used as catalyst for H2 generation [73] orhumidity sensors [74–76]. Nevertheless, in the last decades the interest in MnWO4

relies on its magnetic and electric properties and the interplay between them.

Even if it grows all over the world, it is a rather rare mineral and it usuallygrows embedded in pegmatites and high temperature quartz veins [Fig. 3.1(a)],hardly ever does it alone. Therefore, many methods have been developed to grow

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30 CHAPTER 3. MN1´XCOXWO4

β a

b

c

(a) (b)

(c) (d)

c

a

c

b

FIGURE 3.1: (a) A naturally grown MnWO4 crystal found in Peru and photographed byCarles Millan. (b) Crystal (one of those used in our studies) grown in the laboratory.(c) Projection along the a axis of the crystal structure. Blue circles correspond to tungstenatoms, red ones to oxygen atoms and green ones to manganese. This color code will bemaintained all along the manuscript. (d) Projection along the b axis where the Mn-Wlayered structure along a axis is visible.

crystalline samples of MnWO4 in the laboratory [76–81].

The first recorded investigation in this compound dates back to 1957 [77],when Simanov and Kurshakova published the synthesis and X-ray experimentson MnWO4. The crystal structure of Hübnerite has the symmetry P2/c (No. 13,b-unique axis, standard setting) and it consists of alternative layers of Mn andW atoms along the [100] direction. Oxygen atoms form very distorted octahedraaround Mn and W, and each octahedron shares two edges with the nearest ones,constructing zig-zag chains along the c axis [see Figs. 3.1(c) and 3.1(d)] [77, 82]. Thecrystallographic parameters (cell parameters and atomic positions) are gathered inTable 3.1.

In the sixties magnetic properties at low temperature were reported for thefirst time by Van Uitert and co-workers who revealed a magnetic order

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3.1. MNWO4 31

Cell parameters: a = 4.8238(7) Å, b = 5.7504(10) Å, c = 4.9901(8) Å, β = 91.18(1)o

Atoms Wyckoff position x y zMn 2 f 0.6866(9)W 2e 0.1853(6)O1 4g 0.2132(5) 0.1026(5) 0.9394(4)O2 4g 0.2524(4) 0.3707(6) 0.3918(5)

Volume: 138.3906 Å3

TABLE 3.1: The crystallographic parameters that describe the crystal structure of MnWO4

which has P2/c space-group symmetry are taken from Ref. [82].

below « 16 K [83]. The antiferromagnetic nature of the compound was unveiledfrom the behavior of the magnetization as seen in Fig. 3.2(a), where a clear max-imum of the magnetization appears around 16 K. The paramagnetic tempera-ture ΘP was extrapolated from their data, ΘP « -72 K. The magnetic structurewas investigated by Dachs and co-workers [84, 85] who made neutron diffrac-tion, determined the magnetic propagation vector of the commensurate phase at4.2 K [k = (˘ 1

4 , 12 , 1

2 )] and proposed spin arrangements to describe the magneticstructure. This magnetic structure was supposed to be the unique one belowthe Neél temperature. Nevertheless, specific heat measurements performed byC. P. Landee and E. R. Westrum [86] revealed three anomalies at 6.8(1) K, 12.57(5) Kand 13.36(5) K [depicted in Fig. 3.2(b)], which were attributed to three consecutivemagnetic transitions. The magnetic structures were named as AF3 (13.5 K � T � 12.5 K),AF2 12.5 K � T � 6.8 K) and AF1 (T � 6.8 K).

The spin arrangements in AF1, AF2 and AF3 phases were studied in detail byLautenschläger et. al. [87]. Just below the Néel temperature, TN « 13.5 K, momentsorder collinearly along a direction u within the ac plane with a sinusoidally modu-lated amplitude and an incommensurate propagation vector k = (-0.214, 1

2 , 0.457)[AF3 magnetic phase, Fig. 3.3(a)]. The u direction is in the ac plane, makes an angleof 35˝ with the a axis and is the magnetic easy-axis. The AF2 phase appears in therange 12.5 K ă T ă 7.5 K and presents an additional magnetic component along b,the propagation vector being the same. Thus, the magnetic ordering is an ellipticalcycloidal spin structure as depicted in Fig. 3.3(b). Below 7.5 K the system is againcollinear (along u) and commensurate with k = (˘ 1

4 , 12 , 1

2 ) [AF1, Fig. 3.3(c)]. Laut-enschläger et. al. [87] managed to detect all the magnetic transitions by means ofmagnetic measurements done on powder samples [see Fig. 3.2(c)].

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32 CHAPTER 3. MN1´XCOXWO4

6 8 10 12 14 16 18

0

10

20

30

40

Pb (μ

C/m

2 )

Temperature (K)

Hydrostatic pressure 0 GPa 0.27 GPa 1.54 GPa 1.77 GPa

(f)

5 10 15 20-2.0

-1.5

-1.0

-0.5

0.0

0.5

ΔL/

L 0 (10-5

)

Temperature (K)

b

c

a

T1 T

2 T

N (e)

0 100 200 3000.5

1.0

1.5

2.0

2.5 ^

⎥⎥

M (e

mu

g-1)

Temperature (K)

(a)

0 10 20 30 40 500.40

0.45

0.50

0.55

0.60

χ (c

m3 m

ol-1)

Temperature (K)

1.75

2.00

2.25

(c)χ-1

(cm

-3 m

ol1 )

4 8 12 16 200

20

40

60

Pb (μ

C/m

2 )

Temperature (K)

(d)

4 6 8 10 12 14 160

1

2

3

4

5

Cp (c

al K

-1m

ol-1)

Temperature (K)

(b)⊥

FIGURE 3.2: (a) Magnetization versus temperature along two different directions (easyand hard directions) done by Van Uitert et. al. where the antiferromagnetic nature ofthe crystal is reported [83]. (b) Specific heat by C. P. Landee et. al. [86]. (c) Magneticsusceptibility measured on powder samples (˝) and its inverse (‚) adapted from [87].(d) Electric polarization along the b axis measured by Arkenbout et. al. [31]. (e) Thermalexpansion reported in [95]. (f) Effect of hydrostatic pressure on the polarization alongb axis adapted from Ref. [95].

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3.1. MNWO4 33

(a)

(b) (c)

c

a

c

a

c

a

(d)

b

a

FIGURE 3.3: Magnetic structures of MnWO4 determined by Lautenschläger et. al. [87]:(a) the AF3 magnetic structure projected along the b axis, (b) the AF2 magnetic struc-ture projected along the c axis where the rotation of the moments is noticeable, (c) samestructure projected along the b axis and (d) the commensurate AF1 spin arrangement.

The succession of magnetic phases together with the ratio between the Néel

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34 CHAPTER 3. MN1´XCOXWO4

and the paramagnetic temperature, |Θ|TN

" 1, is due to a strong competition andsubtle balance among multiple magnetic interactions (MnWO4 shows a small mag-netic anisotropy which is comparable in energy to the Dzyaloshinskii-Moriya inter-actions, see Ref. [88]), i. e. magnetic frustration of the system, which was demon-strated by Feng and co-workers in 2011 by means of inelastic neutron scattering[89]. Figures 3.4(a) and 3.4(b) show manganese chains along the c axis and the mag-netic interactions studied by Feng et. al., and Table 3.2 summarizes the strength ofthe couplings. According to their results there are five main interaction (J1, J3, J4, J6and J8) that impose antiferromagnetic coupling between the involved atoms whichcan not be all satisfied simultaneously and hence, the structure becomes magnet-ically frustrated. Moreover, it turns out that these couplings implicate intrachainand interchain interactions along the c and a axes. This could explain the incom-mensurability of the propagation vector along these directions. On the other hand,the magnetic exchange along the b axis is rather weak.

Mn-Mn J1 J2 J3 J4 J5 J6

Dist. 3.286 4.398 4.830 4.990 5.760 5.801Interac. -0.42(1) -0.04(1) -0.32(1) -0.26(1) 0.05(1) -0.43(1)

J7 J8 J9 J10 J11

Dist. 5.883 6.496 6.569 6.875 7.013Interac. -0.12(1) 0.02(1) -0.26(1) -0.15(1) 0.02(1)

TABLE 3.2: Interatomic distances and corresponding magnetic interactions calculated byFeng and co-workers [89].

c c

ab

c

b

(a) (b)

FIGURE 3.4: Two projections of the manganese chains and the intrachain and interchainmagnetic interactions between them.

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3.1. MNWO4 35

It is rather frequent that a crystal structure undergoes distortions through amagnetic transition, this is the so-called magnetoelastic effect or spin-lattice cou-pling [90–94]. In the case of MnWO4 this phenomenon is very weak in the sensethat the relative changes detected in the crystallographic axes are of the order ofΔL/L „ 10´5 [95]. These very small changes were observed by thermal expan-sion measurements, which showed changes in the behavior of ΔL/L through thetransitions, the a axis being the most sensitive to the transitions [see Fig. 3.2(e)].

The interest in this material considerably increased recently with the discov-ery of its ferroelectric nature at low temperatures [30, 31, 79, 80, 95–108], speciallybecause this property was shown to be tightly linked to the magnetic order [30, 31].An electric polarization was observed along the b axis in the temperature rangewhere the AF2 magnetic structure exists [see Fig. 3.2(d)], and consequently, itturned out that the cycloidal AF2 structure is a type II multiferroic phase. Eversince then, the research on MnWO4 has been focused on the study of this multifer-roic phase.

Being a magnetically frustrated system, MnWO4 tends to be very sensitive toexternal fields that can unbalance the subtle equilibrium between the magnetic in-teractions, giving rise to complex field-temperature phase diagrams [98, 109, 110].On top of that, the fact that the ferroelectricity is somehow linked to the magneticstructure, any modification of the spin-arrangement could in principle change theorientation of the electric polarization or even erase it. As a consequence, the appli-cation of external fields can control not only the spin ordering but also the electricpolarization (the case of the effect of hydrostatic pressure is depicted in Fig. 3.2(f)which is adapted from Ref. [95]). In this line, the effect of external magnetic fieldhas been thoroughly investigated [31, 98–104, 109]:

˛ Applying a magnetic field along the easy axis (direction u, in the ac plane)induces different magnetic phases depending on the temperature and theintensity of the field [98, 100]. The AF1 phase is suppressed and the multifer-roic AF2 phase is stabilized at low temperatures and low magnetic field.

However, if the field is increased above « 12 T (below 8 K) the crystal under-goes a magnetic transition to a non-polar phase (HF).

Very recently it has been observed that at very high fields along the easyaxis, above « 40 T, the HF phase is erased, the moments re-arrange in a polarstructure where the electric polarization is parallel to the b axis [103]. Thisphase also appears at lower fields but higher temperatures, without goingthrough the HF phase.

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36 CHAPTER 3. MN1´XCOXWO4

˛ If a relatively low magnetic field (about 11 T) is applied along the b axis, anew polar phase appears in a narrow temperature range, 10 K - 11 K, wherethe electric polarization flips from the b axis to a [99].

˛ On the other hand, if the field is applied along the a or the c axis, the behavioris similar to applying the field along the easy axis [100].

3.2 CoWO4

Known as Krasnoselskite, this mineral is isomorphic to MnWO4. Its crystal struc-ture has P2/c (No. 13, b-unique axis, standard setting) symmetry [82]. The struc-ture is depicted in Fig. 3.5 and the parameters that describe the crystal structureare summarized in Table 3.3.

The first reported studies on CoWO4 were done in the sixties [83, 111]. Themagnetic properties were investigated by Van Uitert [83] who showed that thiscompound is antiferromagnetic below « 55 K and that its paramagnetic tempera-ture ΘP is « -85 K. Its magnetization as the temperature decreases exhibits a clearmaximum at « 55 K [Fig. 3.6(a)] which can be attributed to the Néel temperature.Neutron-diffraction experiments on powder and single crystal samples concludethat moments get ordered collinearly in the ac plane making « 45˝ angle withthe c axis as represented in Fig. 3.5(c) [112–116]. Figure 3.6(b) shows the evolu-tion of the ordered magnetic moment measured by neutron diffraction. The mag-netic transitions occurs together with a contraction of the volume of the unit cellas shown in Fig. 3.6(c) observed by Forsyth and co-workers in Ref. [116].

In CoWO4, the strong single-ion anisotropy leads to a scenario where theanisotropic Dzyaloshinskii-Moriya interaction is too weak to cant the the magneticmoments, and therefore, the isotropic superexchange coupling minimizes the en-ergy by aligning the spins along the easy axis [88].

On the contrary to MnWO4, CoWO4 does not show any electric polarization,it is paraelectric.

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3.2. COWO4 37

(a)

(b)

(c)

c

a

c

a

c

b

FIGURE 3.5: CoWO4: projection of the crystal structure (a) along the a axis and (b) alongthe b axis. The color code is the following: Blue and biggest circles correspond to Watoms; smaller, purple circles inside the octahedra to Co and the smallest red atoms areO atoms. (c) Projection along the b axis of the magnetic structure determined at 15 K byForsyth and co-workers [116].

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38 CHAPTER 3. MN1´XCOXWO4

Cell parameters: a = 4.6698(9) Å, b = 5.6873(23) Å, c = 4.9515(17) Å, β = 90.(0)o

Atoms Wyckoff position x y zCo 2 f 0.6712(8)W 2e 0.1773(4)O1 4g 0.2176(3) 0.1080(4) 0.9321(3)O2 4g 0.2540(3) 0.3757(4) 0.3939(3)

Volume: 131.501 Å3

TABLE 3.3: The crystallographic parameters that describe the crystal structure of CoWO4

which has P2/c space-group symmetry (taken from Ref. [82]).

0 20 40 600

1

2

3

4

Co

mom

ent (μ

B)

Temperature (K)0 20 40 60

130.5

131.0

Uni

t Cel

l Vol

ume

(A3 )

Temperature (K)0 100 200 300

0.50

0.75

1.00

1.25^

⎥⎥

M (e

mu

g-1)

Temperature (K)

(a) (b) (c)

FIGURE 3.6: (a) Thermomagnetic curves of CoWO4 reported by Van Uitert et. al.[83].(b) Ordered Co magnetic moment measured by neutron diffraction and (c) contraction ofthe volume of the unit cell with decreasing the temperature below the magnetic transitionobserved by Forsyth and co-workers [116].

3.3 Mn1´xCoxWO4

Like most of the frustrated multiferroic materials, MnWO4 is extremely sensitiveto small perturbations induced by an external agent such as magnetic field or pres-sure, and also to chemical substitution. One of the main challenges is to enlargethe temperature range where the multiferroic phase is stable. Thus, the objectiveis to modify the magnetic interactions between the atoms so that the multiferroicstructure is favored. Chemical doping is the most promising method for such aim.

MnWO4 has been doped with isovalent metal and nonmetal ions [117–131].In 1969 Weitzel doped the material with iron for the first time and obtained thephase diagram for the Mn1´xFexWO4 solid solution, the so called Wolframites[117]. In 2003 García-Matres and co-workers observed that the incommensuratemagnetic structures disappear when substituting 16% of manganese by iron [118].

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3.3. MN1´XCOXWO4 39

It turned out that the transition temperature to the commensurate AF1 increases.But with further doping this transition temperature decreases and another incom-mensurate phase is promoted which coexists with AF4 (collinear phase of CoWO4).Nevertheless, the higher the doping is the higher the Néel temperature becomes.Some iron doped crystals have been studied under magnetic field. Ye et. al. andChaudhury et. al. [121, 122] reported that applying a magnetic field along the easyaxis stabilizes the multiferroic phase at low doping.

Other groups have tested nonmagnetic atoms as dopants and studied howthe magnetic interactions are modified [123, 124]. It was found that Zn and Mg aregood candidates to stabilize the multiferroic phase, although the Néel temperaturerapidly decreases from « 13 K to « 7 K when 40 % of manganese is substitutedby Zn. These studies demonstrated that the results obtained using nonmagneticatoms were, in overall, independent of the dopant.

Very recently it was discovered that doping with Co2+ ions is particularlyinteresting since it strongly stabilizes the multiferroic phase at low temperaturesand the transition temperatures do not decrease dramatically. According to pre-liminary data on powder samples by Song et. al. [125], the incommensurate mul-tiferroic AF2 phase is expected to substitute the commensurate AF1 ordering atlow temperature for cobalt doping above 3%. At higher doping the AF2 phaseis supposed to transform into AF2’ magnetic phase having the same propagationvector but with different orientation of the moments. The proposed phase diagramis shown in Fig. 3.7. However the magnetic structures of the different compoundswere not unambiguously determined from this work.

This was the state of the research in Mn1´xCoxWO4 at the beginning of thethesis. Parallel to our work, other groups have also actively investigated this solidsolution [125–131].

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40 CHAPTER 3. MN1´XCOXWO4

0.0 0.1 0.2 0.30

5

10

15

20

25

30

AF4

AF2 AF2’+AF4

AF1

Tem

pera

ture

(K)

x in Mn1-xCoxWO4

AF3

FIGURE 3.7: x-T phase diagram obtained by Song and co-workers on powder samples[125]. The empty circles were obtained from the temperature derivative of the suscepti-bility and the filled squares are neutron diffraction. The magnetic phases were unclear inthe dashed area centered at x = 0.10.

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Chapter 4Experimental techniques

All the experimental techniques employed during the thesis are summa-rized in this chapter and a brief description of the instruments used forperforming the experiments is also provided.

4.1 Diffraction

Even if technological advances allow the scientific community to investigate ma-terials by means of a large variety of methods, it is still very difficult to see directlythe atoms in a crystal. To see them we use indirect methods such as diffraction. Byanalyzing the diffraction pattern obtained by the interaction between a wave andthe ordered atoms in the crystal, it is possible to reconstruct the structure of thatcrystalline material.

This technique was first studied in 1665 by Francesco Maria Grimaldi [132]who named this phenomenon as diffringere that in Latin means "break into pieces",referring to the scattering of the light when it founds an obstacle on its way [133].This phenomenon occurs when any wave goes through an obstacle whose dimen-sion is similar to the wavelength of the incoming wave. When the obstacles arelocated periodically, an interference phenomenon enters into play. The interfer-ence pattern is connected to the periodicity of the diffraction grating. In 1913,William Henry Bragg and his son William Lawrence Bragg deduced that if thediffraction grating is a crystal (three dimensional periodic structure), the condition

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42 CHAPTER 4. EXPERIMENTAL TECHNIQUES

of constructive interference is given by the Bragg’s law:

nλ = 2d sin(θ) (4.1)

n being an integer number that corresponds to the order of the diffracted beam, λ

the wavelength of the radiation, d the distance between the planes in the crystaland θ the angle of the diffracted wave with respect to the incoming one.

Within the Born approximation, the elastic scattering by a target of an inci-dent wave vector ki into a final state characterized by k f is given by the differentialcross section:

dΩ(kiσ

1i , k f σ1

f ) =

(m

2πh2

)2 ˇ ă k f σ1f | V(r) | kiσ

1i ą ˇ2 (4.2)

where V(r) is the potential felt by the beam (electrons, neutrons or X-rays) at r inthe field of the scatterer. The σ parameter refers to any other property characteristicof each type of radiation. The potential describes the kind of interaction betweenthe wave and the target. In the next sections neutron and X-ray diffraction will bediscussed.

4.1.1 Neutron Diffraction

Neutrons have properties that make them unique for our purposes. Its mass givesto the neutron, once thermalized, a de Broglie wavelength comparable to the in-teratomic distances, which is a crucial condition for diffraction. Besides these, itis a neutral particle and, therefore, it does not interact with the electrons of thetarget but with its nuclei. Moreover, it can deeply penetrate in the sample giv-ing information about the whole volume and not only about its surface. And, lastbut not least, this particle has a magnetic moment that undergoes a dipole-dipoleinteraction with the magnetic moments of the unpaired electrons in the material.Then, neutrons are suitable not only to study the crystal structure of crystallinematerials but also their magnetic structure. In fact, up to now, they are a uniquetool to study magnetic structures, even if other techniques such as Resonant X-RayMagnetic Scattering provide details of these structures [134–136].

In the case of neutrons, σi and σf are the initial and final spin of the neutrons(σ is the Pauli spin matrix). The scattering due to the crystalline structure is ruledby the interaction of the neutrons and the nuclei of the atoms, which is assumed tobe a δ-function,

Vn(r) =2πh2

m

ÿl,j

bjδ(r ´ Rl j) (4.3)

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4.1. DIFFRACTION 43

where bj is the scattering length1. The intensity of the reflections are proportionalto

dσn

dΩ=

(2π)3

VN

ÿH

|Fn(Q)|2δ(Q ´ H) (4.4)

where Rl j is the position of the atom j in the l unit cell, N is the number of unitcells, V is the volume, Q = k f ´ ki and H is a vector of the reciprocal lattice.The expression is nonzero when H = Q, and this is nothing else but the Braggcondition for reflection.2 Fn is the nuclear structure factor and can be written as

Fn(Q) =ÿ

j

bjeiQ¨rj eWj (4.5)

where eWj is the Debye-Waller factor which takes into account the thermal dis-placements around the equilibrium positions.

One of the advantages of using neutrons is, as mentioned before, that theydo not interact with the electron cloud of the atoms but with the nuclei, by the scat-tering length b. This means that Mn2+ and Co2+ cations are easily distinguishablewith neutron, since bMn2+ = ´3.75 fm and bCo2+ = 2.49 fm [137], whereas it ismore difficult with X-rays.

4.1.1.1 Magnetic diffraction

In the case of magnetic scattering, the expression of the potential is more com-plicated. The magnetic moment of the neutron, μn, undergoes a dipole-dipoleinteraction with field created by the unpaired electrons of the atoms in the crystal:

Vm(r) = ´μn ¨ B(r) = ´μn ¨Nÿ

i=1

(∇ ˆ

(μi ˆ (r ´ Ri)

|r ´ Ri|3)

´ 2μBpi ˆ (r ´ Ri)

h|r ´ Ri|3)

(4.6)

where B is the magnetic field arising from the unpaired electrons, μn is the mag-netic moment of the neutron, μi is the moment of the unpaired electron, and pi,its momentum. With such a potential the scattering cross section from a magneticcrystal for unpolarized neutrons is the following:

dσm

dΩ(Q) = N

(2π)3

V

ÿH

ÿk

|FMK|2δ(Q ´ H ´ k) (4.7)

1It depends on the isotope of the atom.2Coherent elastic scattering occurs when the scattering vector coincides with a reciprocal lattice

vector.

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44 CHAPTER 4. EXPERIMENTAL TECHNIQUES

where k here is the propagation vector of the magnetic structure, and FM is themagnetic structure factor. Note that only the component of the magnetic structurefactor perpendicular to the scattering vector contributes to the magnetic scattering.Hence the intensity of the magnetic reflections are proportional to the square of FM.

FM(Q = k + H) = p f (Q)ÿ

j

Sk,jeiQ¨rj eWj (4.8)

p represents the scattering amplitude at Q=0 for a single magnetic moment of 1μB

and f (Q)9 1Q is the magnetic form factor. Sk,j is the Fourier component of the

magnetic moment distribution.

Indeed, assuming that the magnetic atom j in the unit cell l carries a magneticmoment one can define a magnetic vector, ml j, for each of these atoms that givesthe amplitude and the direction of each magnetic moment. This magnetic momentdistribution can be Fourier expanded according to:

ml j =ÿk

Skj e´2πik¨Rl (4.9)

where Skj is the Fourier component of the distribution associated to the propaga-tion vector k.

The magnetic diffraction may occur at the same position of the nuclear peaks(if k = 0) or at different places in the reciprocal space (if k � 0), see equation (4.7).

Regarding the magnetic neutron-data refinements, except in Chapter 7, themagnetic structures have been studied based on the following expression, whichis the expansion of equation (4.9) for a given magnetic propagation vector k (andthe associated ´k):

ml j(k) = �j(m)uj cos[2π(k ¨ Rl + ϕj)] +�j(m)vj sin[2π(k ¨ Rl + ϕj)], (4.10)

where ml j is the magnetic moment of the atom j in the unit cell l, Rl is the vec-tor joining the arbitrary origin to the origin of unit cell l, and ϕj is a magneticphase. The absolute value of each ϕj is completely arbitrary in the case of an in-commensurate propagation vector. However, the difference of phases betweenthe different Bravais sublattices in the unit cell is important. In the isostructuralMn1´xCoxWO4 compositions there are two magnetic atoms in the unit cell, lo-cated at Mnj=1 ( 1

2 , y, 14 ) and Mnj=2 at ( 1

2 , 1´y, 24 ). In what follows Δϕj stands for

the difference ϕ(Mn2)´ϕ(Mn1). This phase factor is given by representation anal-ysis as developed in Ref. [62] and in Appendix B, and is equal to kz

2 , kz being thecomponent of the propagation vector along c˚.

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4.1. DIFFRACTION 45

For the description of the magnetic structures we use an orthonormal xyzcoordinate system: x coincides with the crystallographic a axis, y belongs to theab plane and z is perpendicular to the ab plane, along c˚. Therefore, in our case,x = a, y = b and z = x ˆ y � c. The unitary vectors u and v are describedin spherical coordinates: θ is the angle with the z = c˚ axis and φ the one withrespect to the x axis of the projection in the xy plane.

Note that collinear structures are described by a unique direction, i. e. �j(m) = 0.

4.1.1.2 Single-crystal diffraction versus powder diffraction

The main difference between powder diffraction and single-crystal diffraction,comes from the fact that for a powder pattern all the reciprocal space is projectedonto one dimension as a function of 2θ. This means that all intensities located atthe same 2θ angle contribute to a unique peak and consequently, the intensity isan average of all those peaks. Due to this averaging, neutron-powder-diffractionexperiments do not provide in certain cases enough information to completely re-solve the spin arrangement of magnetic structures. In a single crystal experimenteach reflection is collected individually.

To illustrate the importance of having a single crystal to determine a mag-netic structure, lets assume that the propagation vector associated to the magneticstructure is k = (0.21, 1

2 , 0), that the moments lay on the ab plane (so does the mag-netic structure factor) and that the diffraction pattern of our material is the oneshown in Fig. 4.1. Two magnetic reflections appear with the scattering vectors Q1

= (1.21, - 12 , 0) and Q2 = (1.21, 1

2 , 0). According to equation (4.7), only the perpendic-ular component of the magnetic structure factor to the scattering vector contributesto the intensity of each reflection. The green line of the plot represents the directionof the magnetic moments in the ab plane and α and β correspond to the angles thatthe scattering vectors make with the magnetic structure factors. As α � β the in-tensity of those reflection is different. However, both reflections, are located at thesame 2θ and, therefore, in powder diffraction they would be added up in a uniquepeak. In that situation, it might occur that the analysis of the data could convergeto different arrangements with equal values for the agreement factors.

Nevertheless, making the experiments on powder samples has certain ad-vantages, such as the observation of all the reciprocal space in one shot, which isof great help when the propagation vector is not known. Actually, this is the mainreason why the determination of the magnetic structures is usually first done on

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46 CHAPTER 4. EXPERIMENTAL TECHNIQUES

β

Q1= (1-10)+k Q2= (100)+k

k = (0.21 0.5 0)

b*

a*

α

FIGURE 4.1: Sketch of a reciprocal space. Nuclear reflections are represented by grey dotsand magnetic reflections by pink dots. For simplicity only two magnetic reflections areplotted. The green line stands for the direction of the magnetic moments. The scatteringvectors, Q1 and Q2, make α and β angle with the direction of the moments, respectively.

polycrystalline samples. Moreover, obtaining a good single crystal, big enough torun a neutron experiment, is often not a trivial task.

In conclusion, the investigation of magnetic phase diagrams usually beginswith the study of polycrystalline material. Once the propagation vectors are knownand if the details of the magnetic structures are unclear, one should try to studysingle crystals. On the contrary, the effect of external fields on the behavior of thematerials must be analyzed on single crystals, in order to apply the field in onefixed direction.

4.1.1.3 Larmor diffraction

This technique, which was developed by Rekveldt [138–140] 10 years ago, is basedon the fact that a spin precesses around a magnetic field with a frequency equal tothe Larmor one:

ωL = ´γB = ´ eg2m

B (4.11)

where B is the magnitude of the magnetic field, g is the g-factor.

Within the classical model, lets assume that neutrons of velocity v and po-larization σx enter a L1 coil, hit the sample and go through another coil L2. Theneutrons spend a certain time in each coil, with a magnetic field perpendicular to

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4.1. DIFFRACTION 47

θ

L

G

αkpe

rp,1

G

θ

L

k per

p=G

/2

k per

p,2

(a) (b)

FIGURE 4.2: Larmor precession sketch.

their velocity and precess an angle

φ =ωLLiviK

(4.12)

in each coil, where i = 1, 2 corresponds to the different coils.

Now, let us imagine the following set up. Neutrons go through a magneticfield, hit the sample and are diffracted according to the Bragg’s law,Q = 2k sin θBu = H, where H is a reciprocal lattice vector normal to the diffract-ing planes. In Fig. 4.2(a) the boundaries of the magnetic field are placed parallelto the diffracting planes, hence, neutrons that satisfy the diffraction condition willundergo an identical Larmor precession while they go through the same magneticfield, because the perpendicular component of the wave vector (or velocity) will beunchanged. If the boundaries of the magnetic field are not parallel to the diffract-ing planes, the path that neutrons will go through in each coil will not be the sameanymore, and the precession of the neutrons will vary in each coil. The final pre-cession angle can be expressed in terms of the cell parameters

φ =4mhH

ωLL(

cos α

cos2 α ´ cot2 θB sin2 α

)9aL, (4.13)

where α is the tilting between the planes and the magnetic field boundary. Bymeasuring the phase difference, variations of the lattice constants (due to thermalexpansion/contraction for example) can be measured very accurately.

4.1.2 Synchrotron X-ray diffraction

Synchrotron X-ray radiation is produced by bending the trajectory of a relativisticcharged particle. It is created artificially in synchrotrons through magnetic fieldsby means of bending magnets, undulators and wigglers. This radiation has certain

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48 CHAPTER 4. EXPERIMENTAL TECHNIQUES

characteristics that makes it unique for many different kind of experiments. It’scharacteristics are the following: white light, very high flux, high brilliance, highstability, polarized beam and pulsed time structure. The high level of intrinsiccollimation and high intensity of the source facilitates the construction of powderdiffractometers with unrivaled resolution as ID31 at the ESRF. Taking profit of thepulsed time structure of the beam, experiments can be performed with good timeresolution, even if this can not combined with ultrahigh resolution. Moreover, thebeam can be tuned to use very shorts wavelengths, which allows to collect dataat very high Q and at the same time to reduce the absorption in the crystal. Thenature of the light provides diffractions patterns with very narrow peak-widths, ofless than 0.05˝, that can be modelled easily.

On the other hand, X-rays do not interact with the nucleus of the atoms butwith the electron cloud that surrounds it. Using the proper potential function inequation (4.2), the nuclear structure factor of a crystal can be written as:

Fn(Q) =ÿ

j

f jeiQ¨rj eWj (4.14)

where fj is the form factor of each j atom and eWj is the Debye-Waller factor whichtakes into account the thermal displacements around the equilibrium positions.The form factor is the Fourier transform of the electron density, which is spreadwithin a cloud, and therefore the Fourier transform depends on Q: it decreases asQ increases. These means that the intensity drops down at high Q.

4.1.3 Refinements and agreement factors

All diffraction data (crystal and magnetic) have been treated using Fullprof Suitepackage [141], except those treated by the superspace formalism in Chapter 7 forwhich the program JANA2006 [61] was used. The crystal and magnetic struc-tures are refined/determined by minimizing the weighted squared difference be-tween the observed data and calculated pattern. Depending on the initial data set,whether it is a powder profile or a set of integrated intensities, the quality of theagreement is measured by different parameters [141–143].

In the case of a powder pattern, where at each 2θi the are yi counts and therefinement of p parameters yields with a model for which the counts of each 2θi isyc,i:

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4.1. DIFFRACTION 49

˛ Profile Factor:

Rp = 100

ři=1,...,n |yi ´ yc,i|ř

i=1,...,n |yi| (4.15)

˛ Weighted Profile Factor:

Rwp = 100

[ři=1,...,n[wi|yi ´ yc,i|2]ř

i=1,...,n wiy2i

] 12

(4.16)

where the weight, wi, is equal to 1σi

being σi the statistical error of yi.

˛ Expected Weighted Factor: this is the expected value for Rwp consideringthat the profile function is perfect.

Rexp = 100

[n ´ př

i=1,...,n wiy2i

] 12

(4.17)

begin n the total number of observed points, i. e. the amount of yi; and p thenumber of refined parameters.

˛ χ2: this value should tend to one.

χ2 =

[Rwp

Rexp

]2(4.18)

This parameter is subject of many systematic errors, such as the underestima-tion of the standard deviations or large collection times with small steps thatyield with small values of Rexp and therefore χ2 results much bigger than 1.

In the cases where the diffraction information is recorded as integrated in-tensities, where F2

obs are the observed integrated intensities, F2calc are the calculated

ones, n runs over the observations from 1 to Nobs, wn is the weight , M the func-

tion to be optimized, M =ř

n wn

(F2

obs,n ´ řk F2

calc,k

)2, k runs over the reflections

that contribute to the observation n and p the number of refined parameters, theagreement factors given to measure the quality of the fitting are the following:

˛ RF2

RF2 = 100

řn[|F2

obs,n ´ řk F2

calc,k|]řn F2

obs,n(4.19)

˛ RwF2

RwF2 = 100

dMř

n wnF2obs,n

(4.20)

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50 CHAPTER 4. EXPERIMENTAL TECHNIQUES

˛ RF2

RF2 = 100

řn[|Fobs,n ´

břk F2

calc,k|]řn Fobs,n

(4.21)

˛ χ2

χ2 =M

Nobs ´ p(4.22)

Very often, the best way to determine the goodness of a proposed model is toobserve the observed intensities versus the calculated ones and to ensure that theresult is physically acceptable.

4.1.4 Instrumentation

For diffraction experiments four different types of instruments were used: single-crystal neutron-diffractometers, a thermal-neutron spectrometer and a synchrotron-X-ray powder-diffractometer. Most of the neutron-diffraction experiments havebeen performed on D23 and therefore, a detailed description of this instrument isprovided to the reader in the following.

D23: it is CEA/CRG a thermal-neutron two-axis diffractometer for single crystals[144]. It is installed on the thermal-neutron guide H25 at the Institut LaueLangevin (ILL) in Grenoble, France. The neutron guide is equipped withsupermirrors and it is curved in such a way that no direct view of the reactor-core is possible: this means that there is not fast-neutron contamination andthus background levels are very low. D23 has been designed to work with orwithout polarized neutrons, in the incident wavelength range 1 Å - 3 Å.

As neutrons enter the instrument coming from the neutron-guide, they en-counter either of the two monochromators that provide different wavelengths:a vertical focused pyrolythic graphite (PG) 002 (in reflection) with λ « 1.3 Åor a flat Cu 200 (in transmission) that provides λ « 2.4 Å. In the secondaryshielding, another monochromator (a Heusler 111 in transmission) can bemounted when neutrons need to be polarized. The crystal is installed onthe sample-table. The scattered neutrons are counted by a lifting detectormounted in an arc. The sketch of the instrument is shown in Fig. 4.3.

The diffractometer operates in normal-beam geometry which means that ithas three angular motions: ω, γ and ν. The first corresponds to the rotation ofthe sample-table around the vertical axis and γ and ν are the angular motions

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4.1. DIFFRACTION 51

Sample table

Sample Detector Beamstop Optical lineMonochromator 2

(Heusler)Monochromator 1

(PG or Cu)

FIGURE 4.3: Sketch of the single-crystal thermal-neutron diffractometer D23. The figurehas been adapted from [144].

of the detector, the rotation in the equatorial plane and out of the horizontalplane, respectively. The limitations on the angular motions restrain the max-imum value of the Bragg angle (2θB « 130˝). The lifting detector can cover asymmetric angular range ´29˝ ă ν ă 29˝.

This instrument can hold different environments such as pressure cells,cryostats and cryomangets. The mechanics of the instrument is completelynon magnetic which enables the instrument to support high field magnets.However the use of these sample environments introduces certain experi-mental restrictions. The following restrictions concern the cryomagnets usedin this thesis:

(i) Smaller angular range: The magnets restrict the accessible ν range andcan even reduce the reachable reflections to a single plane. The acces-sible ν angular range depends on each magnet, ´5˝ ă ν ă 20˝ for the6T-CNRS magnet and ´3˝ ă ν ă 10˝ for the 12T-CEA-CRG magnet.

(ii) Blind regions: The mechanical structure of the magnet has columnswhich considerably reduce the neutron flux and hence, they produce

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52 CHAPTER 4. EXPERIMENTAL TECHNIQUES

blind areas from the experimental point of view.

(a)

(b)

-70 -60 -50 -40 -30 -20 -10 01.5

2.0

2.5

3.0

3.5

Neu

tron

coun

ts (x

104 )

ω(o)

FIGURE 4.4: (a) Sketch of the 6T-CNRS vertical-field magnet that can be used on D23.(b) ω-scan of the direct beam which shows the influence of the pillars on the intensity ofthe beam.

D15: it was a thermal-neutron diffractometer for single crystals located at the ILL.It was installed on an inclined beam tube (IH4). Three wavelengths could

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4.1. DIFFRACTION 53

be set up, λ «0.85 Å, λ «1.17 Åand λ «1.54 Å. The instrument could beoperated in either four-circle or normal-beam mode.

6T2: This diffractometer is a thermal-neutron four-circle diffractometer with lift-ing counter situated at the Laboratoire Léon Brillouin (LLB) at Saclay, France[146]. It is equipped with two vertically focusing monochromators: Cu 220with λ « 0.90 Å and an Er filter and a PG 002 with λ « 1.55 Å and 2.35 Å(PG filter). Two types of geometries can be used: four-circle geometry withan Eulerian cradle for structural studies of large unit cells (cell volumes ofmore than 1000 Å) and high resolution studies (phase transitions, etc . . . )or a lifting-counter geometry when using cryomagnet, dilution cryostat andhigh pressure cell for magnetic studies (normal-beam geometry).

IN22: this instrument is a three-axis spectrometer installed at the end position ofthe thermal supermiror guide H25 at the ILL [145]. It is equipped for fullpolarization analysis. The Heusler (111) provides a high flux of polarizedneutrons. The sample, analyzer and detector are mounted on nonmagneticmodules with adjustable distances between them.

ID31: this beamline is devoted to high resolution powder-diffraction at the Euro-pean Synchrotron Radiation Facility (ESRF) in Grenoble, France [147]. Threeundulators produce X-rays with wavelengths 2.48 Åto 0.21 Å. A double-crystal monochromator is used to choose a unique λ: Si 111 crystals (forstandard operation) or Si 311 crystals (for higher energy resolution). ID31works with spinning capillaries or flat plate specimens. Nine detectors mea-sure the diffracted intensity as a function of 2θ. Each detector is precededby a Si 111 analyzer crystal. The flux at the sample is 1.5 ˆ 1012 photonsmm´2 s´1 at 0.43 Åwavelength and 6.1 ˆ 1012 photons mm´2 s´1 at 0.85 Å.It can hold many sample environments: liquid-helium-cooled cryostat, Ox-ford Cryosystems Cryostream cold-nitrogen-gas blower, hot-air blower, mir-ror furnace, . . .

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54C

HA

PTER4.

EXPERIM

ENTA

LTEC

HN

IQU

ES

Samplebeing glued in a pin

D15

Samplemounted on D15

FIGURE 4.5: Pictures of single crystals of Mn1´xCoxWO4. Some crystals appear unglued, one being glued and the last one already prepared onD15 ready for a neutron diffraction experiment.

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4.2. BULK MAGNETIC MEASUREMENTS 55

Neutron-diffraction experiments were usually performed on big samples,which usually were not oriented. Thus, very often, the orientation was done ona neutron diffractometer. Usually big crystals are recommended because of thefact that neutrons hardly see materials and therefore, increasing the size of thesample helps to gain intensity. On the other hand, having a high quality single-crystal proper for neutron diffraction is sometimes rather difficult: often one findsreflections that are split in several reflections. It occurs if the crystal is slightly bro-ken for example. The orientation of the pieces is a bit different and consequently,each piece diffracts in a different direction. So, even if for electric and magneticmeasurements this misalignment is not important, it makes a neutron-diffractionexperiment unfeasible. Coming upon a suitable crystal for a neutron-diffractionexperiment requires testing a lot of crystals. Once a good crystal is found, usu-ally it has to be oriented and glued in a pin. Figure 4.5 shows several pictures ofunglued and glued crystals.

Once single-crystal data was collected the absorption in the sample was cor-rected using the Cambridge Crystallography Subroutine Library [148].

For synchrotron-X-ray powder-diffraction experiments, λ = 0.40000(1) Åwavelength was selected. Powder samples were obtained from crushed singlecrystals and were enclosed in a borosilicate-glass capillary (with 0.5 mm �), anda good powder averaging was ensured by appropriate spinning of the capillaryin the beam. The counting time was about 1.5 hours to have the desired statisticsover the angular range 4˝ to 42˝ in 2θ.

The crystal and magnetic structures have been produced using program VESTA[149] and FullProf Studio [141].

4.2 Bulk magnetic measurements

The magnetization is defined as the magnetic moment of the material per unit vol-ume (or per unit mass) and the susceptibility is defined as a proportionality betweenthe magnetization and the applied magnetic field (it indicates the degree of mag-netization of a material in response to an applied magnetic field).

When a dc field, H, is applied to a sample it gets magnetized and this state ischaracterized with a certain magnetization, M and magnetic susceptibility, χ, val-ues. If an ac field is applied or superimposed to the dc field, the magnetic response

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56 CHAPTER 4. EXPERIMENTAL TECHNIQUES

becomes time dependent. The fundamental difference between dc and ac measure-ments is that the former provides discrete information about the magnetizationcurve whereas the latter gives information about the slope of the magnetizationcurve at a certain point H. The ac susceptibility is given by

χac =dMdH

. (4.23)

At low frequencies, where the measurement is more similar to dc magnetometry,the magnetic moment of the sample follows the M(H) curve that would be mea-sured in a dc experiment. As long as the ac field is small, the induced ac magneti-zation is

Mac =dMdH

¨ hac, (4.24)

where hac is the time dependent applied magnetic field, and ω is the driving fre-quency:

hac = h cos ωt. (4.25)

Changing the applied dc-magnetic-field, different parts of the M(H) curvecan be accessed. Since the ac measurement is sensitive to the slope of M(H) andnot to the absolute value, small magnetic changes can be detected even when theabsolute magnetization is large.

At higher frequencies, the ac magnetization of the material does not followthe dc magnetization curve due to dynamic effects in the sample,

Mac = M cos (ωt ´ θ) (4.26)

θ being the phase shift between the field and the magnetization. For this rea-son, the ac susceptibility is often known as dynamic susceptibility. In this case,the magnetization of the sample may lag behind the driving field. Thus, the ac-magnetic-susceptibility measurement yields two quantities: the magnitude of thesusceptibility, χ, and the phase shift. Alternately, one can think of the susceptibil-ity as having an in-phase, or real, component χ1 and an out-of-phase, or imaginary,component χ2.

χac = χ1 cos (ωt) + χ2 sin (ωt) (4.27)

where,

χ1 = χ cos θ χ2 = χ sin θ θ = arctanχ2χ1 χ =

bχ12 + χ22

The imaginary component, χ2, indicates dissipative processes in the sample.If there are any time-dependent relaxation processes in the system, the induced

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4.2. BULK MAGNETIC MEASUREMENTS 57

magnetization cannot follow the field instantaneously and this leads to a non-zeroθ and, therefore, to a non zero χ2. The imaginary component of the susceptibility,is also non zero if there are any non linearities or hysteresis in the dc magneticresponse of the material. Besides that, both χ1 and χ2 are very sensitive to thermo-dynamic phase changes, and are often used to measure transition temperatures.

Both dc- and ac-susceptibility measurements were used to obtain the mag-netic phase-transition temperatures and the type of transitions of the handled ma-terials during the thesis.

Superconducting quantum interference device (SQUID) magnetometer,a widely used instrument for magnetic measurements nowdays, is able to detectextremely tiny magnetic fields. This device is based on the tunneling of super-conducting electrons across a very narrow insulating gap (known as a Josephsontunnelling junction) between two superconductors. Its high sensitivity is possiblebecause it responds to changes of magnetic field associated with one flux quantum.The working principle of a SQUID is based on the possibility to convert magneticflux into an electrical voltage.

In principle, a measurement is performed in the SQUID magnetometer byfirst moving the sample along the symmetry axis of superconducting detectioncoils and a magnet. Due to its movement, the magnetic moment of the sampleinduces an electric current in the detection coils. A change of magnetic flux in thesecoils changes the persistent current in the detection circuit. Hence, the change ofthe current in the circuit produces variation of output voltage in the SQUID, whichis essentially proportional to the magnetic moment of the sample.

The detection coils are basically a single superconducting wire in the form ofthree counterwound coils configured as a second-order gradiometer. This configu-ration eliminates spurious signals caused by the fluctuations of the large magneticfield from the superconducting magnet, and also reduces noise from nearby mag-netic objects in the surrounding environment.

Two commercial instruments from Quantum Design, Superconducting Quan-tum Interference Device (SQUID) and Physical Property Measurement System (PPMS),were used to measure bulk magnetic characteristics, namely ac-susceptibility anddc-susceptibility and magnetization. While the ASMS ac-susceptometer/dc-magnetomeroption of the PPMS was used to obtain all ac-susceptibility and most of the magne-tization data, the SQUID (model MPMS_LX) was used for low field magnetizationmeasurements, to benefit from the specific characteristics of the two instruments:

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58 CHAPTER 4. EXPERIMENTAL TECHNIQUES

˛ The superior sensitivity of the RSO3 option of the SQUID « 5 ˆ 10´8 emu(« 5 ˆ 10´5 emu for the PPMS dc option).

˛ The possibility to get rid of the remnant field in the SQUID superconduct-ing magnet by quenching it and also demagnetizing the shield screening thestray fields. No quenching option is provided in the PPMS and one couldonly minimize the remnant field of the type II superconducting magnet byperiodically cycling the field when reducing it amplitude to zero.

˛ The PPMS maximum field of 9 T (7 T for the SQUID) and the possibilityto study ac-susceptibility up to such high dc fields. Furthermore since thePPMS works as a dc-magnetometer based on the extraction technique, themeasurements could be executed rather fast and also the maximum signalthat could be measured is much bigger that the 0.3 emu upper limit for theRSO option of the SQUID.

The samples were generally mounted in clear plastic drinking straws withinfew degrees in respect of the magnetic field direction [see Fig. 4.6]. The specificconditions at which the data have been collected will be specified when presentingthe results for each crystal.

FIGURE 4.6: Single crystal located in a plastic straw. The red pen is provided as scale.

3When the sample is moved up and down it produces an alternating magnetic flux in the pick-upcoil which leads to an alternating output voltage of the SQUID device. By locking the frequency of thereadout to the frequency of the movement (RSO, reciprocating sample oscillation), the magnetometersystem can achieve the extremely high sensitivity for ultra small magnetic signals as described above.

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4.3. PYROELECTRIC MEASUREMENTS 59

4.3 Pyroelectric measurements

Electric polarization measurements were kindly provided by Prof. A. A. Mukhin.The measurements of the electric polarization as a function of the temperaturewere performed by detecting the pyroelectric current by a Keithley 6517A elec-trometer. Samples were prepared with electrodes sputtered on opposite faces ofthe crystaland for different crystallographic orientations. In order to ensure a sin-gle domain ferroelectric state a poling field Ep up to ˘3000 V/cm was first appliedin the paraelectric state, prior to cooling the sample down the lowest temperature,usually 1.9 K. At this point, the voltage was set to zero.

Since polarization measurements are concerned with the bulk polarizationwithin the sample, it was important to remove the effect of surface charges whichhad built up due to the application of an electric field. A bleed resistor box wasbuilt for this purpose, which dissipated the surface charge when the voltage sup-ply was disconnected. The sample was then connected to a Keithley 6517A elec-trometer, and the temperature of the sample was increased whilst measuring thepyroelectric current. The temperature dependence of the pyroelectric current typ-ically shows a finite signal in the ferroelectric phase, with a sharp anomaly at TFE.Above TFE, the signal becomes flat, showing that electric polarization no longerexists in the system. In order to convert the data into a measure of the polarization(in Cm´2), the following relationship is used:

P =

żIdtA

(4.28)

Data are integrated with respect to time. This integration method ensures that zeropolarization is calculated in the temperature range where the pyroelectric currentis zero, i.e. above TFE. The integral of the current is divided by the sample area(electrode area) to give the polarization, which can be plotted against temperature.

4.4 Crystal synthesis

I would like to stress that I did not participate in the synthsis of the crystals. Theywere all grown by Prof A. M. Balbashov from Moscow Power Engineering Instituteand kindly provided to us by Prof. A. A. Mukhin. The single crystals were grownby a floating zone method using an FZ apparatus URN-2- ZM [150]. The crystalgrowth was performed in an air atmosphere at a linear speed of 8-10 mm/h witha counter-rotation of crystal and feed rod « 20 min´1. Finally the growing crystal

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60 CHAPTER 4. EXPERIMENTAL TECHNIQUES

was annealed at T = 1100 ˝C. Small pieces of the crystal were cut for oriented byX-ray diffraction for bulk magnetic and electric polarization measurements.

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Chapter 5Interest and Objectives

Taking into consideration these previous work, it was worth studying the effect ofdoping the prototype multiferroic MnWO4 with cobalt on single crystals. Experi-ments on powder samples evidenced that the phase diagram of this solid solutionis very rich and interesting, however single crystal experiments were lacking tofirmly establish the magnetic structures through the phase diagram.

The two main points that make this doping so interesting are the following:

˛ The multiferroic phase is stabilized when doping above 3%.

˛ The transition temperature to the multiferroic phase does not decrease dras-tically.

The principal objective of this thesis was to contribute to the field of multifer-roics by studying this highly frustrated family. Namely, we would like to shedlight on the following points:

(i) The effect of substituting Mn2+ by the anisotropic Co2+ ion on the magneticfrustration.

(ii) The interplay between the magnetic structure and the induced electric polar-ization.

(iii) The existence of new multiferroic phases varying the cobalt content: the mag-netic structures and the orientation of the induced electric polarization. The

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62 CHAPTER 5. INTEREST AND OBJECTIVES

ultimate aim was to complete the x ´ T phase diagram of the Mn1´xCoxWO4

family of compounds.

(iv) The spin-lattice coupling, specially at the multiferroic transition.

(v) Magnetic-field-induced transitions.

(vi) Magnetic field effects, depending on the direction along which it is applied,on the magnetic structures and on the electric polarization: H ´ T phase dia-grams.

(vii) How the magnetic and electric phase transitions occur.

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Part II

Magnetic, ferroelectric andstructural properties of

Mn1´xCoxWO4 under zeromagnetic field

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Chapter 6Crystal Structure of the

Mn1´xCoxWO4 family

The study of the crystal structure of the Mn1´xCoxWO4 family are presentedin this chapter, where the influence of both, the temperature and the cobaltcontent, are investigated.

The study of the interplay between magnetic transitions and the crystal struc-ture is crucial for the investigation of multiferroic materials. In fact, the structuralchanges across the magnetic transitions are usually the clue to ascribe the origin ofthe concomitant appearance of electric polarization. Nevertheless, in some materi-als, those structural variations are so small that their detection is very difficult. Thisis usually the case of the magnetic oxides that belong to the type II multiferroicswhere the inverse Dzyaloshinskii-Moriya interaction induces small displacementsof oxygen atoms that are located between magnetic atoms, producing a chargedistribution that yields a macroscopic electric polarization. MnWO4 being a proto-type of type II multiferroics, is of high interest to search for structural changes thatoccur at its magnetic transitions, specially at the multiferroic transition.

With this aim, the crystal structure of all the compositions were determinedby means of single-crystal neutron-diffraction. Note that with X-ray diffractionthere could be some problems to locate light atoms beside heavy ones, that is thecase of oxygen atoms close to manganese/cobalt and tungsten atoms (X-rays basi-cally see the electron cloud and not the nuclei, and small atoms such as oxygen canbe shadowed by bigger atoms). The study done on the pure compound at different

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66 CHAPTER 6. CRYSTAL STRUCTURE OF THE MN1´XCOXWO4 FAMILY

temperatures allows us to observe the spin-lattice coupling, whereas the analysisof the crystal structure of the distinct species of the series will permit us to estab-lish a relationship between the structural changes induced by the cobalt and thedistinct magnetic structures in the phase diagram.

A careful study of the cell parameters of all compositions has been done atroom temperature by means of high resolution synchrotron-X-ray powder-diffractionbecause this technique provides very accurate values for the cell parameters. Allthe samples were measured in the same conditions (environment, wavelength, . . . )and therefore, the systematic errors that those factors could introduce in the deter-mination of the cell parameters is the same. This allows a precise comparison andto see very small relative differences with high precision.

6.1 The crystal structure of MnWO4 across the mag-netic transitions investigated by single-crystalneutron-diffraction

The crystalline structure of the pure MnWO4 was carefully studied by means ofneutron-diffraction experiments on the D15 diffractometer at the Institute Laue-Langevin (ILL) in Grenoble operating in a four-circle geometry.

Crystal structures were refined at T = 2 K, 5 K, 8 K (magnetically orderedstate) and T = 16 K (paramagnetic state). The cell parameters (a, b, c and β) wereobtained at these temperatures from the centering of 63 strong reflections and arelisted in Table 6.1. No significant changes in the cell parameters were observed.

In addition to the cell variation, modifications in the crystallographic struc-ture may occur at magnetic transitions, i. e. atomic displacements linked to the ori-entation of the moments. To check this, 290 independent reflections [symmetry av-eraged from 818 reflections with Rint(16 K) = 1.21%, Rint(8 K) = 1.21%,Rint(5 K) = 1.24% and Rint(2 K) = 1.30%] were collected at each temperature. Fromthe refinements it was not possible to see any difference in the structure at thosetemperatures. Figure 6.1 shows the nuclear structure as refined, where blue cir-cles represent tungsten atoms, green ones are manganese atoms and red circlescorrespond to oxygen atoms. This color code will be maintained throughout themanuscript. The agreement plots of the refinements at each temperature are alsogiven in Fig. 6.1. Table 6.1 summarizes the refined atomic positions which are the

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6.1. THE CRYSTAL STRUCTURE OF MNWO4 ACROSS THE MAGNETIC TRANSITIONS INVESTIGATEDBY SINGLE-CRYSTAL NEUTRON-DIFFRACTION 67

Temperature T = 16 K T = 8 K T = 5 K T = 2 KMagnetic phase Paramaganetic AF2 AF2-AF1 AF1

a/Å 4.824(2) 4.823(2) 4.824(2) 4.824(2)b/Å 5.755(3) 5.755(3) 5.757(2) 5.756(2)c/Å 5.002(2) 5.001(3) 5.004(2) 5.004(2)β/˝ 91.08(3) 91.10(4) 91.09(2) 91.06(3)Volume/Å3 138.83(3) 138.79(4) 138.98(2) 138.95(3)

Mn y 0.6841(2) 0.6841(2) 0.6842(2) 0.6841(2)Biso/Å2 0.27(2) 0.27(2) 0.26(3) 0.26(3)

W y 0.1803(1) 0.1802(1) 0.1803(1) 0.1802(1)Biso/Å2 0.17(2) 0.17(1) 0.17(2) 0.16(2)

O1 x 0.2108(1) 0.2108(1) 0.2108(1) 0.2108(1)y 0.1023(1) 0.1024(1) 0.1024(1) 0.1024(1)z 0.9423(1) 0.9422(1) 0.9423(1) 0.9422(1)

Biso/Å2 0.29(2) 0.29(1) 0.29(1) 0.29(1)

O2 x 0.2510(1) 0.2511(1) 0.2511(1) 0.2511(1)y 0.3746(1) 0.3746(1) 0.3746(1) 0.3746(1)z 0.3933(1) 0.3933(1) 0.3933(1) 0.3933(1)

Biso/Å2 0.33(2) 0.32(1) 0.32(1) 0.32(1)

RF/% 3.04 3.03 3.10 3.06RF2/% 4.46 4.47 4.60 4.64RF2w/% 6.13 5.79 5.90 5.92χ2 55.2 59.8 61.9 62.7

TABLE 6.1: The crystal-structure parameters of MnWO4 refined at 16 K, 8 K, 5 K and 2 Kfrom single-crystal neutron-diffraction. Atomic coordinates are given relative to the cell.All the experiments were performed in a four-circle geometry on D15.

same, within the errors bars, for all the temperatures.

The distances between Mn-O atoms and the angles between O-Mn-O atomsthat form the octahedra are summarized in Table 6.2 and Table 6.3 respectively,and the sketch of the octahedra is plotted in Fig. 6.2. Within the accuracy of theexperiment, there is no clear evidence of any structural change in MnWO4. Theoctahedra are very distorted, but this distortion is not influenced by the magneticordering.

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68 CHAPTER 6. CRYSTAL STRUCTURE OF THE MN1´XCOXWO4 FAMILY

0

2

4

6

8

0 2 4 6 8

F2obs(arb. units)

F2 calc(a

rb. u

nits

)

T = 2 K

RF= 3.06 %RF2W = 5.92 %

0 2 4 6 80

2

4

6

8

RF= 3.03 %RF2W = 5.79 %

T = 8 K

F2 calc(a

rb. u

nits

)F2

obs(arb. units)

0 2 4 6 8

0

2

4

6

8

RF= 3.10 %RF2W = 5.90 %

F2 ca

lc(a

rb. u

nits

)

F2obs(arb. units)

T = 5 K

0 2 4 6 80

2

4

6

8

RF= 3.04 %RF2W = 6.13 %

F2 ca

lc(a

rb. u

nits

)

T = 16 K

F2obs(arb. units)

c

ab

FIGURE 6.1: Nuclear structure at 16 K, 8 K, 5 K and 2 K in the pure MnWO4 compoundtogether with the calculated versus observed integrated-intensities from the refinements.

MMMMMMMMMMMMMMMMMMMMMMMnnnnn1111111

Mn2

O1

OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO1111111111111111111OOOOOOOOOOOOOOOOOOOOOOOOOOOOO222222222222222

OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO222222222222222222222222222222222222

O3

O33c

ab

FIGURE 6.2: Sketch of the distorted octahedra. Oi and Oii are equivalent in the octahedra.The atoms are labeled as in Tables 6.2 and 6.3.

Temp.Angle/˝

O1-Mn1-O11 O1-Mn1-O2 O1-Mn1-O3 O2-Mn1-O22 O2-Mn1-O3 O3-Mn1-O33

T = 16 K 161.98(5) 96.08(5) 81.31(5) 108.56(5) 87.41(5) 76.74(5)T = 8 K 161.97(4) 96.09(4) 81.31(4) 108.58(4) 87.41(4) 76.73(4)T = 5 K 161.97(5) 96.10(5) 81.32(5) 108.59(5) 87.40(5) 76.72(5)T = 2 K 161.98(5) 96.09(5) 81.32(5) 108.57(5) 87.42(5) 76.72(5)

TABLE 6.3: Angle between O-Mn-O at distinct temperatures in MnWO4.

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6.1. THE CRYSTAL STRUCTURE OF MNWO4 ACROSS THE MAGNETIC TRANSITIONS INVESTIGATEDBY SINGLE-CRYSTAL NEUTRON-DIFFRACTION 69

TemperatureDistance/Å

Mn1-O1 Mn1-O2 Mn1-O3 Mn1-Mn2 <Mn-O>

T = 16 K 2.1584(5) 2.1054(8) 2.2717(9) 3.2780(9) 2.1785(4)T = 8 K 2.1576(5) 2.1050(8) 2.2720(9) 3.2781(9) 2.1782(4)T = 5 K 2.1591(5) 2.1057(8) 2.2728(8) 3.2799(9) 2.1792(4)T = 2 K 2.1594(5) 2.1052(8) 2.2723(9) 3.2795(9) 2.1790(4)

TABLE 6.2: Interatomic distance between manganese and oxygen atoms and the averagedistance for the Mn-O bond in MnWO4.

6.1.1 Larmor-diffraction study of the spin-lattice coupling

Chaudhury and co-workers [95] found a weak but non-zero spin-lattice couplingby means of thermal-expansion measurements, see Fig. 6.3. According to theirresults, a and c parameters decrease across the paramagnetic-to-AF3 transition atT1 « TN , whereas the monoclinic axis, b, increases. At the AF3-to-AF2 magnetictransition, where a new magnetic component arises along the b axis [87], no evi-dence of any change in the behavior of the cell parameters was seen. However, atthe AF2-to-AF3 (T2) a sharp change of the evolution of a, b and c was found. The aand b parameters undergo a sharp decrease, whereas c suffers an abrupt increase.The overall evolution of the cell parameters from 20 K to 4 K yields with slightlyhigher value of b and smaller c and a parameters, a being the one that presents thebiggest change. So, with the aim of detecting such small changes of the cell param-eters, we turned to a spin-echo experiment in the Larmor-diffraction configuration.

The Larmor diffraction is a neutron-diffraction technique very sensitive to therelative changes of the cell parameters, Δl/l „ (10´5 ´ 10´6). We had the oppor-tunity to perform a test experiment on IN22 with the help from N. Martin andL. P. Regnault. The crystal was mounted on the triple-axis IN22 in the Larmor-diffraction configuration. The obtained variations of the a and b parameters areshown in Fig. 6.4 (c and β parameters have not been investigated during this exper-iment). b was recorded when decreasing temperature, whereas a was measured onincreasing. The variation of a and b is very small (Δa/a „ 10´5 and Δb/b „ 10´6),however clear changes in the evolution of these parameters can be observed atthe temperatures which coincide with the magnetic transitions: T1 « TN andT2 « TAF2´to´AF1. The observed evolution of both parameters agrees well withthe evolution reported in Ref. [95]: whilst temperature decreases, b slightly in-creases and a decreases, the relative change of b being much smaller. Though thereis a discrepancy with the work of Chaudhury et. al. concerning the absolute change

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70 CHAPTER 6. CRYSTAL STRUCTURE OF THE MN1´XCOXWO4 FAMILY

4 8 12 16 20-2.0

-1.5

-1.0

-0.5

0.0

0.5

ΔL/

L 0 (10-5

)

Temperature (K)

b

c

a

T2 T1

FIGURE 6.3: Thermal expansion of a, b and c parameters observed by Chaudhury andco-workers in Ref. [95].

of the parameters. Nevertheless, this small change of the cell parameters confirmsthe fact that there is a small magnetoelastic coupling and that conventional experi-ments of single-crystal neutron-diffraction, like the one performed on D15, are notsensitive enough to detect such effects in this compound.

2 4 6 8 10 12 14 16 18

-3

-2

-1

0

T2 ~ 5.2 K

Δa/

a (x

10-5

)

Temperature (K)

T1 ~ 13.5 K

-10

-5

0

Δb/

b (x

10-6

)

FIGURE 6.4: Relative variation of a and b parameters observed by Larmor diffraction.

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6.2. INFLUENCE OF THE CO ON THE CRYSTAL STRUCTURE OF THE MN1´XCOXWO4 FAMILY (X =0.05, 0.10, 0.15 AND 0.20) 71

6.2 Influence of the Co on the crystal structure ofthe Mn1´xCoxWO4 family (x = 0.05, 0.10, 0.15and 0.20)

Substituting Mn2+ by Co2+ induces changes in the magnetic behavior of the ma-terial [125]: the AF1 phase disappears and the multiferroic AF2 phase takes itsplace (at x « 3%). With increasing the amount of cobalt up to about x = 10%,the multiferroic phase undergoes distortions which results in a different orienta-tion of the electric polarization [126]. When the amount of cobalt is around 15%coexisting magnetic phases arise. Some phases are only observed in these com-position [127]. At higher doping the coexistence of phases disappears [125]. Themagnetic structures of all this compositions will be discussed in Chapter 8. Seeingsuch differences on the magnetic behavior depending on the cobalt concentration,it is important to make a careful study of the crystal structures and find out the dif-ferences between the compositions that could modify the magnetic interactions.

To search for variations of the nuclear structure of the crystals investigatedduring the thesis, the nuclear structures of all them have been checked by single-crystal neutron-diffraction. All the compositions are isoestructural to the pureMnWO4, with space group P2/c. All the structural parameters are summarized inTable 6.4. The difference between the neutron scattering-lengths of Mn (-3.75 fm)and Co (2.49 fm) permitted us to refine of the concentrations of these two speciesin all the compositions. There are some parameters that were not refined when theexperimental environments used for some experiments did not allow to collect aproper set of reflections for that task. Those parameters are marked by a star (*)in Table 6.4. In the case of the 5% Co-doped composition, several experimentswere needed to completely refine the structure. However, the agreement factorsdemonstrate that the quality of the refinements in all cases was good enough.

Even if there are not significant variations of the relative positions of theatoms in the crystallographic unit cell, the absolute vales of the distances variynoticeably with the Co concentration. Tables 6.5 and 6.6 contain the Mn-O dis-tances and the O-Mn-O bond angles, respectively. Note that Oi and Oii are equiv-alent in the octahedra. This information is given only for x = 0.05, 0.10 and 0.20due to the fact that the 15% Co-doped composition was not completely refinedbecause of the experimental conditions1. All the compositions were not investi-

1x = 0.15 was studied inside a magnet. The angular aperture did not allow to collect reflectionswith l � 0 and thus none of the parameters related to this direction could be refined.

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72 CHAPTER 6. CRYSTAL STRUCTURE OF THE MN1´XCOXWO4 FAMILY

Mn0.95Co0.05WO4 Mn0.90Co0.10WO4 Mn0.85Co0.15WO4 Mn0.80Co0.20WO4

Temperature 2 K 17 K 2 K 25 KInstrument 6T2 D23 D23 D23Orientation a and b axis c axis c axis a axisλ/Å 0.900 1.2832 1.2773 1.2773

a/Å 4.816(7) 4.8068(9) 4.7982(7) 4.784(2)b/Å 5.751(2) 5.744(1) 5.742(1) 5.731(2)c/Å 4.997(2) 4.983(5) 4.98* 4.976(2)β/˝ 90.99(7) 90.91(5) 90.8* 90.80(7)Volume/Å3 138.3906 137.5633 136.337* 137.2635

Mn y 0.6839(7) 0.6847(2) 0.6860(9) 0.6847(5)Occ. 96.0(1)% 90.6(2)% 85(1)% 82.0(6)%Biso 0.5(2) 0.20(3) 0.36(5) 0.31(7)

Co y 0.6839(7) 0.6847(2) 0.6860(9) 0.6847(5)Occ. 4.0(1)% 9.4(2)% 15(1)% 18.0(6)%Biso 0.5(2) 0.20(3) 0.36(5) 0.31(7)

W y 0.1799(6) 0.1802(1) 0.1809(5) 0.1807(3)Biso 0.4(1) 0.16(1) 0.20(6) 0.08(3)

O1 x 0.2111(5) 0.21142(9) 0.2115(4) 0.2124(8)y 0.1022(4) 0.10290(7) 0.1026(3) 0.1031(1)z 0.9416(4) 0.9410(4) 0.9410* 0.9401(2)

Biso 0.5(1) 0.29(1) 0.40(5) 0.24(2)

O2 x 0.2510(5) 0.25150(9) 0.2515(4) 0.2522(8)y 0.3747(4) 0.37507(7) 0.3751(3) 0.3752(1)z 0.3933(4) 0.3941(4) 0.3939* 0.3928(2)

Biso 0.5(1) 0.29(1) 0.40(5) 0.24(2)

RF/% 5.99 and 6.59 2.39 4.44 3.65RF2/% 8.12 and 8.78 3.09 6.45 5.65RF2w/% 3.91 and 4.99 4.99 9.63 6.78χ2 25.7 and 29.3 11.2 97.3 52.0

TABLE 6.4: Structural parameters of the Mn1´xCoxWO4 family(x = 0.05, 0.10, 0.15 and 0.20). All the experiments were performed in normal-beamgeometry. Those parameters were used for the refinement of the magnetic structures.

gated at the same temperature, but the study of the pure MnWO4 showed us thatno drastic changes were occurring between 2 K and 25 K. The nuclear structureof the 10% Co-doped material was check at different temperature below 17 K andindeed, it did not suffer any modification within the experimental resolution. Theresults show that the distances between atoms shrink with increasing the amountof cobalt, which turns in a smaller volume of the octahedra. The Mn-O-Mn angleincreases, however the Mn-Mn interatomic distance becomes smaller at high Codoping. The angles between the oxygens in the basal plane of the octahedra alsoundergo small variations: the angle between O2-Mn1-O3 grows at expenses of the

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6.2. INFLUENCE OF THE CO ON THE CRYSTAL STRUCTURE OF THE MN1´XCOXWO4 FAMILY (X =0.05, 0.10, 0.15 AND 0.20) 73

one between O2-Mn1-O33. The modifications of the octahedra do not affect theirorientation with respect to the c axis. If one draws a plane that contains the oxygenatoms in the equatorial plane of the octahedra, and another one that contains theapical ones, see Fig. 6.5, the orientation of those planes does not vary within theexperimental resolution. The equatorial plane of the octahedra is about 134.5˝ farfrom c and the apical plane is basically perpendicular to it.

ConcentrationDistance/Å

Mn1-O1 Mn1-O2 Mn1-O3 Mn1-Mn2 <Mn-O>

x = 0% 2.1584(5) 2.1054(8) 2.2717(9) 3.2780(9) 2.1785(4)x = 5% 2.157(2) 2.101(4) 2.267(5) 3.274(5) 2.175(2)

x = 10% 2.150(2) 2.089(1) 2.265(2) 3.272(1) 2.168(1)x = 20% 2.146(2) 2.074(3) 2.255(3) 3.267(3) 2.159(2)

TABLE 6.5: Interatomic distances between manganese and oxygen atoms and the aver-age distance for the Mn-O bond in the x = 0.05, 0.10 and 0.20 compositions. The purecompound has been included for making easier the comparisons.

Concen.Angle/˝

O1-Mn1-O11 O1-Mn1-O2 O1-Mn1-O3 O2-Mn1-O22 O2-Mn1-O3 O3-Mn1-O33 Mn1-O3-Mn2

x = 0% 161.98(5) 96.08(5) 81.31(5) 108.56(5) 87.41(5) 76.74(5) 95.42(5)x = 5% 162.00(9) 96.0(1) 81.30(9) 108.3(1) 87.5(1) 76.74(9) 95.4(1)x = 10% 161.6(1) 96.02(8) 81.22(4) 108.5(1) 87.52(9) 76.57(7) 95.62(7)x = 20% 161.58(4) 96.2(1) 81.34(8) 108.22(9) 87.8(1) 76.31(1) 95.8(2)

TABLE 6.6: Angle between O-Mn-O in x = 0.05, 0.10 and 0.20 compositions. x = 0 hasbeen included to complete the view of the effect of the Co.

6.2.1 Influence of the Co doping on the crystal unit-cell

To obtain the cell parameters with better accuracy, all the compositions were in-vestigated at room temperature by means of high resolution synchrotron-X-raypowder-diffraction, at the ultra-high resolution ID31 diffractometer of the Euro-pean Synchrotron Radiation Facility (ESRF) in Grenoble, France. Figure 6.6 showsthe evolution of the cell parameters a, b, c and β as a function of the cobalt concen-tration, gathered in Table 6.7. The corresponding diffraction pattern and the fitsare depicted in Fig. 6.7. The sample with 15 % of cobalt contained very little impu-rity, which most likely comes from the crystal surface that was scratched to obtainthe powder. This explains why a small range in 2θ has been erased for the profilematching and why the reliability factors are slightly worse for this composition

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74 CHAPTER 6. CRYSTAL STRUCTURE OF THE MN1´XCOXWO4 FAMILY

c

ab

FIGURE 6.5: Sketch of the planes used to calculate the orientation of the octahedra: pinkplane correspond to the (001) plane, the blue one contains the apical oxygens of the octa-hedra and the orange one the oxygens in the equatorial plane of the octahedra.

than for the rest.

As expected from Vegard’s law [151], the lattice parameters decrease as theconcentration of cobalt increases, because Co2+ ions are smaller in size than Mn2+;the ionic radii of Co2+ ions with coordination six is 0.745 Å, whereas that of Mn2+

ions in the same configuration is 0.83 Å (both in high spin state) [152]. Thus, the de-crease of the cell parameters is linear, a being the one which decreases more rapidly,ΔaΔx « ´0.16 Å/x%Co. By fitting the data, the cell parameters of the complete familycould be predicted. In particular, the values predicted for the pure CoWO4 agreevery well with the ones given in Table 3.3. Indeed, the cell parameters for thepure CoWO4 extrapolated from the linear fits are the following: a « 4.669(1) Å,b « 5.690(1) Å, c « 4.950(1) Åand β « 90.05(1)˝, that should be compared to theobserved values (a = 4.6698(9) Å, b = 5.6873(23) Å, c = 4.9515(17) Å, β = 90.(0)˝measured by Weitzel in Ref. [82]).

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6.3. SUMMARY AND CONCLUSIONS 75

MnWO4 Mn0.95Co0.05WO4 Mn0.90Co0.10WO4 Mn0.85Co0.15WO4 Mn0.80Co0.20WO4

a/Å 4.83242(1) 4.82445(1) 4.81621(1) 4.80785(2) 4.79984(2)b/Å 5.76211(1) 5.75871(1) 5.75494(1) 5.75174(2) 5.74768(2)c/Å 4.99986(1) 4.99751(1) 4.99481(1) 4.99256(2) 4.99000(2)β/˝ 91.1514(1) 91.0970(1) 91.0418(1) 90.9900(2) 90.9300(2)Volume/Å3 139.193(1) 138.818(1) 138.418(1) 138.042(1) 137.646(1)

Rp/% 7.61 7.11 7.08 12.8 7.72Rwp /% 9.79 9.36 9.37 17.4 9.87Rexp/% 3.61 3.37 4.09 4.91 3.28χ2 7.34 7.69 5.25 12.5 9.48

TABLE 6.7: Cell parameters of the Mn1´xCoxWO4 family (x = 0, 0.05, 0.10, 0.15 and 0.20)determined by profile matching of powder-diffraction patterns obtained on ID31 at roomtemperature.

4.8004.8054.8104.8154.8204.8254.8304.835

a = 4.83250(1) - 0.16318(9) x

a (A

)

5.748

5.752

5.756

5.760

5.764

b = 5.76217(1) - 0.07156(9) x

b (A

)

0 5 10 15 204.990

4.992

4.994

4.996

4.998

5.000

x % Co

c (A

)

(a) (b)

(c) (d)

β = 91.1518(1) - 1.100(1) x0 5 10 15 20

90.95

91.00

91.05

91.10

91.15

c = 4.99988(1) - 0.04949(7) x

β (o )

x % Co

FIGURE 6.6: Evolution of the cell parameters with the amount of Co, x. In panels (a), (b),(c) and (d) are given the evolution of a, b, c and β, respectively. Empty circles correspondto the profile-matching results (errors inside the symbols) and the grey line is a linearfitting.

6.3 Summary and conclusions

To summarize, the substitution of Mn2+ by Co2+ reduces the volume of the unitcell, which in turn provokes the contraction of the oxygen-made octahedra but not

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76 CHAPTER 6. CRYSTAL STRUCTURE OF THE MN1´XCOXWO4 FAMILY

any tilt of them. The length of Mn-O and Mn-Mn bonds become shorter and theangle between O2-Mn1-O3 gets bigger at expenses of the one between O2-Mn1-O33,see Fig. 6.2 and Tables 6.5 and 6.6. The result that there is not systematic rotationof the MO6 octahedra with increasing the cobalt content is an important one. Thisfact allows us to conclude that not only the average basal plane is not systemati-cally rotating but, much more important, locally the specific basal plane of a givenoctahedra can be considered constant with increasing the Co content.

Given the complexity of this system (not less than ten exchange couplinginteractions competing) it is not possible to establish a direct relationship betweenthe evolution of the structure with x and the magnetic properties. Nevertheless,these small but clear modifications of the atom positions should be at the origin ofthe cobalt-induced changes in the magnetic and polar phase diagram, as it will bediscussed further in the thesis.

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6.3. SUMMARY AND CONCLUSIONS 77

4 6 8 10 12 14 16 18 20

Mn0.90Co0.10WO4

Mn0.95Co0.05WO4

Inte

nsity

(arb

.uni

ts)

Fobs

Fcalc

Fobs

-Fcalc

Bragg peak

MnWO4(a)

(b)

(c)

(d)

(e)

Mn0.85Co0.15WO4

Mn0.80Co0.20WO4

2θ(ο)

FIGURE 6.7: High resolution X-ray powder-diffraction patterns fitted by the profile-matching method. Data were collected at room temperature.

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78 CHAPTER 6. CRYSTAL STRUCTURE OF THE MN1´XCOXWO4 FAMILY

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Chapter 7Detailed description of the

magnetic structures of MnWO4

The truth of the story lies in the details.

Paul Auster (1947 - )

I n this chapter, the incommensurate magnetic structures of MnWO4 are ana-lyzed using the superspace-symmetry formalism.

The magnetic structures of the multiferroic MnWO4, see Chapter 3, havebeen previously analyzed using the representation analysis method [87] and theLandau theory of phase transition [107, 108]. In this chapter, the incommensuratestructures of MnWO4 are re-determined, ascribing to each phase its correspondingsuperspace group, that determine the possibility of having electric polarization ornot and hence, multiferroicity.

7.1 Symmetry analysis

Since only first order magnetic reflections have been observed in pure MnWO4,one can describe its magnetic modulation with a single harmonic. Hence, themagnetic moments of a magnetic atom μ in the position rμ, inside the cell l of

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80 CHAPTER 7. DETAILED DESCRIPTION OF THE MAGNETIC STRUCTURES OF MNWO4

the crystal, can be expressed as follows:

Mμ,l = Mμe´2πik¨(l+rμ) + Mμe2πik¨(l+rμ) (7.1)

The introduction of the superspace formalism essentially reduces to:

(i) The introduction of a modulation function for each magnetic atom μ, definedalong a continuous coordinate x4, which is directly related to the previousexpression:

Mμ(x4) = Mμe´2πix4 + Mμe2πix4 (7.2)

This simply means that the value of the magnetic moment Mμ,l of the atomμ at any unit cell l is given by the value of the corresponding Mμ(x4) modu-lation function for x4 = k ¨ (l + rμ).

(ii) The definition of a set of symmetry operations (the superspace group) that ingeneral constrain the possible form of these modulation functions and cor-relate the modulation functions of symmetry related atoms within the unitcell, such that only some of them are independent. In this way, an asymmet-ric unit for both atomic positions and modulation functions can be defined.The symmetry operations tR, θ|t, τu of the relevant superspace group (θ = -1or +1 labels the inclusion or not of time reversal respectively) are ordinaryspace group operations tR, θ|tu belonging to the grey space group1 of theparamagnetic phase, but followed by some specific translation related withτ along the internal coordinate x4 of all atomic modulation functions.

In the simplest case of a purely incommensurate propagation vector (seeRef. [66] for the general expressions or the summary given in Chapter 2 Section2.6.3), the symmetry relation between the magnetic modulation functions of twoatoms related by a superspace symmetry operation tR, θ|t, τu is given by :

Mμ(RI x4 + τ) = θ det(R)R ¨ Mν(x4) (7.3)

with the average position of atom μ being related with that of atom ν by the spacegroup operation tR|tu : tR|turν = rμ + l. Here, RI is +1 or ´1 if R keeps k in-variant or transforms k into ´k, respectively. For μ = ν, equation (7.3) defines a

1Magnetic space groups combine the crystallographic space-groups with the time reversal operator.If an symmetry operator contains the time inversion, θ = ´1, it is known as black and if it does not,θ = +1, it is named white, a grey space group combines all spatial operator with both θ = ´1 andθ = +1 doubling the number of operator in the space group.

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7.1. SYMMETRY ANALYSIS 81

constraint on the form of the modulation. Note that the symmetry relations de-fined in the superspace formalism, in contrast with the traditional Bertaut repre-sentation method, include in general rotational or roto-inversion transformationsthat change the sign of the propagation vector. These symmetry transformationshave been previously considered using a so-called "non-conventional use" of theco-representations [153] or within the frame of Landau theory [107, 108]. As ex-plained in Ref. [66], the use of the superspace formalism introduces automaticallyand in a simple form the whole set of symmetry constraints associated to these op-erations, for any degree of freedom of the structure, including non-magnetic ones.

The space group of the paramagnetic phase of MnWO4 is P2/c. In the crys-tallographic unit cell there is one independent manganese atom located at the2 f Wyckoff position: Mn1 at ( 1

2 , y, 14 ). The symmetry related atom Mn2 within

the unit cell is at ( 12 , ´y + 1, 3

4 ). In general, for the symmetry analysis of a mag-netic structure, the parent symmetry has to be taken into account, which in thiscase corresponds to the grey magnetic space-group P2/c11 of the paramagneticphase, where 11 refers to the time inversion operation. This space group con-sists of all the symmetry operations of the ordinary P2/c space group, plus anequal number of operations obtained by multiplying all of them by time reversal{11|0 0 0 0} (which is also a symmetry operation of the paramagnetic phase). Theincommensurate propagation vector, k = (´0.214, 1

2 , 0.457), lies on the symmetryline G (α 1

2 γ) of the Brillouin Zone [154]. The corresponding little group, formedby the symmetry operators that leave k invariant, is Pm11. There are two possi-ble magnetic irreducible representations (irreps) of this little group. They are bothone-dimensional and are defined by the corresponding irreps of the little co-groupm11 (see Table 7.1).

irrep 1 m 11 Superspace group Generators

mG1 1 1 -1 P2/c11(α 12 γ)00s {my|0 0 1

2 0},{1|0 0 0 0}, {11|0 0 0 12 }

mG2 1 -1 -1 P2/c11(α 12 γ)0ss {my|0 0 1

212 }, {1|0 0 0 0}, {11|0 0 0 1

2 }

TABLE 7.1: Irreps of the little co-group, m11, which define the two possible magnetic irrepsof the paramagnetic group P2/c11 with k = (α, 1

2 , γ). The label of the resulting superspacegroup as well as its generators are provided in each case.

As explained in Ref. [66], if the magnetic order is realized according to a sin-gle one dimensional irrep, a unique superspace group can be assigned to the result-ing magnetic structure. All the operations of the little group are kept in this super-space group, but a (0 0 0 1

2 ) translation (along the internal space) has to be addedto those operators that have ´1 character. In addition, all operations transforming

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82 CHAPTER 7. DETAILED DESCRIPTION OF THE MAGNETIC STRUCTURES OF MNWO4

k into ´k are also maintained: its chosen representative (for instance the spaceinversion) can be chosen with zero translation along the internal space by a con-venient choice of the phase of the magnetic modulation, while the remaining op-erations can be obtained from the application of the internal product of the group.Following these rules, the two possible superspace groups corresponding to thetwo possible active irreps at the G line can be derived (the program JANA2006[61] has also an option to derive them automatically). According to the notationusually employed, they can be labeled as P2/c11(α 1

2 γ)00s and P2/c11(α 12 γ)0ss,

and their generators are listed in Table 7.1. It is important to stress that thesesuperspace groups include operations that transform k into ´k. In both cases,the corresponding magnetic point-group is 2/m11, which implies the absence insuch phases of any induced ferroelectricity and of any linear magnetoelectric re-sponse. It is also worth mentioning that the magnetic superspace groups associ-ated to magnetic irreps can also be obtained using the program ISODISTORT [155],but in this case the symmetries are given in standard settings, which are not thesettings used here.

Centered unit cells are used to simplify the description of crystal symmetryin standard crystallography. This can also be done in the case of incommensuratesuperspace symmetry. The propagation vector (α, 1

2 , β) is not purely incommen-surate, as it includes a component 1

2 along y. Because of this commensurate com-ponent, the symmetry relations and constraints of the modulations are given byan equation more complex than equation (7.3) (the symmetry relations and con-strains imposed by P2/c11(α 1

2 γ)00s and P2/c11(α 12 γ)0ss superspaces are devel-

oped in Appendix C). This complication can however be avoided, and equation(7.3) can be applied, by using a unit cell doubled along y, so that the propaga-tion vector in its reciprocal basis becomes k1 = (α, 0, β). The effect of the (0, 1

2 , 0)component of the modulation propagation vector is maintained by the introduc-tion of an appropriate centering translation in superspace. More specifically, oneconsiders a cell a, 2b, c such that now there are four Mn atoms per unit cell.The (0, 1

2 , 0) component of the original modulation only means that the atomswithin this unit cell related by a translation b have modulations with a π phaseshift. This is ensured by including in the superspace group a centering transla-tion (0 1

2 0 12 ). Indeed, according to equation (7.3), if the atoms in the doubled

cell are labeled as Mn1 = (12 , y1

2 , 14 ), Mn2 = (1

2 , ´y1+12 , 3

4 ), Mn11 = (12 , y1+1

2 , 14 ) and

Mn22 = (12 , ´y1+2

2 , 34 ), the modulations of the atoms Mn11 and Mn22 are 1

2 shifted

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7.1. SYMMETRY ANALYSIS 83

with respect to those of the other two atoms:

MMn11(x4) = MMn1(x4 +12 )

MMn22(x4) = MMn2(x4 +12 ).

(7.4)

The description of a superspace group with this optional centered unit cellsis usually indicated in the superspace group label by means of an "X". ThusP2/c11(α 1

2 γ)0ss becomes X2/c11(α0γ)0ss and P2/c11(α 12 γ)00s becomes

X2/c11(α0γ)00s. The symmetry operations of the superspace groups are describedin detail in Table 7.2. The apparent new reflections introduced by the duplicationof the basic unit cell are canceled out due to the centering (0 1

2 0 12 ), as depicted in

Fig. 7.1. In the following, we will use this X-centered description of any relevantsuperspace group.

-k'

-k

(1 1 0 -1)b*/2

k'

k

(1 1 0 1)

(1 0 0 0)

(0 1 0 -1)

(1 2 0 0)(0 2 0 0)

-k'

b*

a*

-k

Incident beam

FIGURE 7.1: Scheme of the indexation (hklm) of diffraction peaks according to a, 2b, csupercell and a propagation vector k1 = (α 0 β) together with a (0 1

2 0 12 ) centering trans-

lation (X-centering). Full circles and grey rhombuses correspond to nuclear and magneticreflections, respectively. Magnetic reflections are projected on the plane (hk0). Empty cir-cles and empty rhombuses represent systematic extinctions caused by the X-centering,which forces that both magnetic and nuclear reflections obey the k + m = 2n reflectioncondition.

A magnetic phase can result from the activation of several primary irrepmodes. In this case, if the propagation vector is incommensurate, its superspace

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84 CHAPTER 7. DETAILED DESCRIPTION OF THE MAGNETIC STRUCTURES OF MNWO4

P2/c11(α 12 γ)0ss

{1|0000} x1 x2 x3 x4 m{2y|001

212 } -x1 x2 -x3+1

2 x2-x4+12 m

{1|0000} -x1 -x2 -x3 -x4 m{my|001

212 } x1 -x2 x3+1

2 -x2+x4+12 m

{1’|00012 } x1 x2 x3 x4+1

2 -m

X2/c11(α0γ)0ss{1|0000} x1 x2 x3 x4 m{2y|001

212 } -x1 x2 -x3+1

2 -x4+12 m

{1|0000} -x1 -x2 -x3 -x4 m{my|001

212 } x1 -x2 x3+1

2 x4+12 m

{1’|00012 } x1 x2 x3 x4+1

2 -m{1|01

2012 } x1 x2+1

2 x3 x4+12 m

P2/c11(α 12 γ)00s

{1|0000} x1 x2 x3 x4 m{2y|001

20} -x1 x2 -x3+12 x2-x4 m

{1|0000} -x1 -x2 -x3 -x4 m{my|001

20} x1 -x2 x3+12 -x2+x4 m

{1’|00012 } x1 x2 x3 x4+1

2 -m

X2/c11(α0γ)00s{1|0000} x1 x2 x3 x4 m{2y|001

20} -x1 x2 -x3+12 -x4 m

{1|0000} -x1 -x2 -x3 -x4 m{my|001

20} x1 -x2 x3+12 x4 m

{1’|00012 } x1 x2 x3 x4+1

2 -m{1|01

2012 } x1 x2+1

2 x3 x4+12 m

TABLE 7.2: Representative operations of the space groups P2/c11(α 12 γ)0ss and

P2/c11(α 12 γ)0ss described in primitive setting and in the X-centered one. Both gener-

alized Seitz-type notation (left column) and symmetry cards as used in JANA2006 arelisted. The labels -m or m indicate whether the operation includes the time inversion(-m) or not (m). The remaining operations are obtained by the internal product of thegroup.

symmetry is described by the intersection of the superspace groups associatedwith each individual irrep. This intersection depends on the relative phases of

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7.1. SYMMETRY ANALYSIS 85

the corresponding modulations. When two primary incommensurate irrep mod-ulations are superimposed, only one of the two modulation phases can be arbi-trarily chosen. This is done by associating to a representative operation of its su-perspace group transforming k into ´k a null translation along the internal space.For the calculation of the intersection of the superspace groups associated withdifferent irrep modulations, one has to consider that if a magnetic modulation isphase shifted by a quantity φ the symmetry operators of its superspace group thatleave invariant k do not change, whereas the operations that transform k into ´kacquire an additional (0 0 0 2φ) translation along the internal space [66]. If we con-sider the superposition of two primary modes transforming according to any of thetwo possible mG1 and mG2 irreps, taking into account all possible relative phaseshifts, several different superspace groups are possible and can be readily calcu-lated following these rules. They are listed in Table 7.3 (Appendix D shows howto build Table 7.3). One can see that the superposition of two modes mG1 or twomodes mG2 reduce the magnetic point-group symmetry to m11, i. e. a polar groupthat allows a spontaneous electric polarization in the ac plane, while the superpo-sition in quadrature of a mode mG1 and a mode mG2 implies a 211 point-groupsymmetry, i. e. a polar symmetry that allows the build-up of electrical polarizationalong b.

Δφ = 0, 12

mG1 + mG1 X2/c11(α0γ)00s 2/m11mG2 + mG2 X2/c11(α0γ)0ss 2/m11mG1 + mG2 X111(α0γ)0s ´111

Δφ = 14 , 3

4

mG1 + mG1 Xc11(α0γ)0s m11mG2 + mG2 Xc11(α0γ)ss m11mG1 + mG2 X211(α0γ)0s 211

Δφ = arbitrarymG1 + mG1 Xc11(α0γ)0s m11mG2 + mG2 Xc11(α0γ)ss m11mG1 + mG2 X111(α0γ)0s 111

TABLE 7.3: Superspace groups resulting from the superposition of two irrep magneticmodulations with a phase shift Δφ between them. The magnetic point-group of eachsuperspace group is in the last column.

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86 CHAPTER 7. DETAILED DESCRIPTION OF THE MAGNETIC STRUCTURES OF MNWO4

7.2 Paraelectric AF3 magnetic phase

If we assume that the paraelectric magnetic structure of the AF3 phase correspondsto a single active irrep modulation, described in Table 7.1, its magnetic symmetryshould be described by either the X2/c11(α0γ)00s or X2/c11(α0γ)0ss superspacegroups. Both symmetries include the inversion operation t1|01

2012u that relates

atoms Mn1 and Mn2. According to equation (7.3) this means that, the magneticmodulations of both atoms are in both cases related in the form:

MMn2(´x4 +12) = MMn1(x4) (7.5)

which implies that only the modulation of one atom, say Mn1, is independent,while the two atoms Mn2 and Mn1 have, in both superspace groups, identicalmodulations but with opposite chirality. The distinction between the two possiblesymmetries comes from the operation t2y|001

2 τu (see Table 7.2), which keeps atomMn1 invariant and therefore constrains the form of its modulation. According toequation (7.3):

MMn1(´x4 + τ) = 2y ¨ MMn1(x4) (7.6)

If we call [M1x(x4), M1y(x4), M1z(x4)] the three components of the magneticmodulation function of Mn1 along the three crystallographic directions, in the firstsuperspace group τ = 0, according to equation (7.6), the modulation of the x, zcomponents of the magnetic moment should be sine-like, while the y componentcosine-like.

τ = 0 Ñ M1α(x4) = Ms1α sin(2πx4) (α = x, z) (7.7)

M1y(x4) = Mc1y cos(2πx4). (7.8)

On the other hand, for X2/c11(α0γ)0ss, τ = 12 and the constraint of sine and cosine

forms are exchanged, so that:

τ =12

Ñ M1α(x4) = Mc1α cos(2πx4) (α = x, z) (7.9)

M1y(x4) = Ms1y sin(2πx4) (7.10)

while according to equation (7.5), the spin modulation of atom Mn2 is given by

τ = 0 Ñ M2α(x4) = Ms1α sin(2πx4) (α = x, z) (7.11)

M2y(x4) = ´Mc1y cos(2πx4) (7.12)

τ =12

Ñ M2α(x4) = ´Mc1α cos(2πx4) (α = x, z) (7.13)

M2y(x4) = Ms1y sin(2πx4) (7.14)

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7.2. PARAELECTRIC AF3 MAGNETIC PHASE 87

Thus, the real amplitudes sets (Ms1x, Mc

1y, Ms1z) or (Mc

1x, Ms1y, Mc

1z) fully describethe magnetic structure.

It is important to stress that, even if the modulation of Mn1 were reducedto the plane ac, the choice of cosine or sine functions for its description does notrepresent a mere phase shift in the global spin wave, because equation (7.5) isalso included in the model, and it implies a completely different choice for thecorrelation with the spins of Mn2 atoms, if one or the other superspace group ischosen. In both alternative symmetries three parameters are required to describethe magnetic structure.

It is worth to compare with the usual representation approach. As it is usu-ally used, the representation analysis only introduces the irrep restrictions corre-sponding to the symmetry operations of the little group, which keeps k invariant.Atoms Mn1 and Mn2 in the unit cell are related in the paramagnetic phase by thespace group operation tmy|011

2u, belonging to the little group. Thus, the represen-tation method introduces a relation between the spin modulations of the two Mnatoms corresponding to this symmetry operation. This relation is equivalent tothe one resulting from the superspace operation tmy|01

212 , 1

2 + τu (X-setting) withτ = 0 for mG1 and τ = 1

2 for mG2, which is present in their respective superspacegroup. However, no restriction equivalent to the one of equation (7.6) is consid-ered, and therefore, in the representation approach, the spin modulation of oneof the atoms is fully free [87]. This means that the relative phases between themodulations of the three spin components are free parameters. As in an incom-mensurate modulation, one of the phases can be fixed arbitrarily, the number ofrefinable parameters of the model is five, compared with only three parameterswhen using the superspace group description. In fact, the model considered inRef. [87], applying the standard representation analysis method, is equivalent tousing the minimal superspace groups Xc11(α0γ)0s or Xc11(α0γ)ss correspondingto the superposition of at least two phase-shifted irrep modes of the same irrepsymmetry (see Table 7.3). Note that this model for the magnetic structure wouldimply a magnetic point-group m11, and therefore polarity within the xz plane. As ithappens in many other studies, the additional restrictions necessary to increase thesymmetry of the model and to reduce the modulations to a single irrep mode weresubsequently introduced in Ref. [87] by means of intuitive arguments based onthe search for simple modulations (collinear, spiral, etc.). In general, the structuresdescribed applying the representation approach (as normally used) are compati-ble with those obtained using the superspace formalism, but less constraints areintroduced, and are less symmetric.

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88 CHAPTER 7. DETAILED DESCRIPTION OF THE MAGNETIC STRUCTURES OF MNWO4

Both superspace groups, X2/c11(α0γ)00s and X2/c11(α0γ)0ss, were tried al-ternatively to refine the magnetic structure from the single crystal data obtainedon D23. The 166 observed magnetic reflections resulted after averaging into 66 in-dependent ones with Rint = 8.67. From the refinement it can be clearly concludedthat the magnetic symmetry of the AF3 phase is X2/c11(α0γ)0ss. The refined pa-rameters that describe the possible magnetic structure with different symmetryas well as the agreement factors are given in Table 7.4 and the structure is de-picted in Fig. 7.2 according to the X2/c11(α0γ)0ss symmetry. The refined valueof the amplitude Ms

y describing the symmetry-allowed sine modulation along they-direction is very small, although larger than its standard deviation. It was thenfixed to zero, and Table 7.4 also shows the result of this second refinement. Thislatter refinement, with the weak Ms

y component neglected, reduces the model toa sinusoidally-modulated collinear magnetic modulation in the ac plane, with thedirection of the moments (the so-called u direction) making approximately a 39(1)˝angle with the a axis and 1.88(4) μB amplitude. This approximate model is directlycomparable and agrees with the magnetic structure proposed in Ref. [87], whereamplitude is 2.1 μB and the angle is 35˝.

c

a

FIGURE 7.2: Projection along b axis of the AF3 phase of MnWO4 at 13 K (Table 7.4). Onlythe non-periodic modulation within a region 4a ˆ 2b ˆ 2c is represented.

It is important to stress that the structure of the AF3 phase is not forced bysymmetry to be collinear. A Hamilton test [156] indicates that the weak non-zerovalue of the amplitude Ms

y is statistically significant. This non-zero modulationfor the y spin component makes the spin waves helical, but of opposite chiralitiesin the two Mn atoms. This situation is represented in Fig. 7.3, where for clarityreasons an artificially large component Ms

y has been introduced in the model. Thetwo modulations are related by space inversion [see equation (7.5)], which forbids

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7.3. AF2 MULTIFERROIC PHASE 89

Ms1x Ms

1y Ms1z Mc

1x Mc1y Mc

1z |mmax| |mmin|X2/c11(α0γ)00s

0.5(2) 0.000* 0.2(2) 0.000* 1.8(1) 0.000* 1.8(1) 0.5(1)RF(obs)= 40.45%, RwF(obs)= 43.64%, RwF2 (obs)= 70.86%

X2/c11(α0γ)0ss0.000* 0.05(3) 0.000* 1.46(3) 0.000* 1.18(2) 1.88(4) 0.05(3)RF(obs)= 7.15%, RwF(obs)= 5.51%, RwF2 (obs)= 10.59%

X2/c11(α0γ)0ss0.000* 0.00** 0.000* 1.47(2) 0.000** 1.17(2) 1.88(3)RF(obs)= 7.23%, RwF(obs)= 5.63%, RwF2 (obs)= 10.86%

* Constrained by symmetry. ** Fixed manually.

TABLE 7.4: Refined parameters (μB) that describe the magnetic structure of MnWO4 at13 K with constrains imposed by the X2/c11(α0γ)00s and X2/c11(α0γ)0ss superspacegroup, the latter including or not a y-component in the magnetic modulation. Maximumand minimum values of the ordered moments are given in the last two columns, i. e. thesemi-major and semi-minor axes of the helix in the first two models and the amplitude ofthe modulation in the second model. Parameters are defined in the text.

any induced net electric polarization.

7.3 AF2 multiferroic phase

The propagation vector does not change at the transition from the AF3 phase tothe multiferroic AF2 phase, but the symmetry is clearly broken, as a spontaneouspolarization appears. In the simplest scenario, one can consider that this symmetrybreaking is due to the activation of an additional irrep magnetic mode. Table 7.3contains all possible superspace groups that can result from the superposition oftwo primary irrep magnetic modes. One can see that there are three magneticsymmetries in the table that could in principle describe a multiferroic structure,as their magnetic point-groups are 211, m11 and 111. Those point-groups allowthe appearance of electric polarization along the b axis, in the xz plane, or alongan arbitrary direction, respectively. It is known that the polarization in MnWO4

arises along the b axis [30, 31, 96]. Hence, according to Table 7.3 the superspacegroup symmetry that must correspond to phase AF2 is X211(α0γ)0s (the symmetryoperations of the space group are described in Table 7.5). The inversion and theglide plane that relate Mn1 and Mn2 in the AF3 phase are therefore lost at thetransition, and these two manganese atoms become symmetry-independent. The

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90 CHAPTER 7. DETAILED DESCRIPTION OF THE MAGNETIC STRUCTURES OF MNWO4

Mn2 Mn1 b

a

Mn22 Mn11

FIGURE 7.3: Projection along c axis of the AF3 phase with an arbitrarily large amplitude ofthe symmetry-allowed y-component of the magnetic modulations. The symmetry of theAF3 phase allows a cycloidal component in the Mn modulations but of opposite chiralityfor Mn1 and Mn2.

symmetry X211(α0γ)0s keeps however the operation t2y|0012

12u 2. This means that

the spin modulations of both atoms Mn1 and Mn2 atoms maintain the symmetryconstraints imposed by equations (7.9) and (7.10), but with amplitudes that are nolonger symmetry-related. There are therefore six parameters to refine, three foreach manganese atom: (Mc

1x, Ms1y, Mc

1z) and (Mc2x, Ms

2y, Mc2z).

M1α(x4) = Mc1α cos(2πx4) (α = x, z) (7.15)

M1y(x4) = Ms1y sin(2πx4) (7.16)

M2α(x4) = Mc2α cos(2πx4) (α = x, z) (7.17)

M2y(x4) = Ms2y sin(2πx4) (7.18)

2The 12 shift of this operation along the internal variable could be made zero by a convenient shift of

the origin of the x4 coordinate, but we prefer to keep the origin considered in the AF3 phase, to allow amore direct comparison with this phase.

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7.3. AF2 MULTIFERROIC PHASE 91

P211(α 12 γ)0s

{1|0000} x1 x2 x3 x4 m{2y|001

212 } -x1 x2 -x3+1

2 x2-x4+12 m

{1’|00012 } x1 x2 x3 x4+1

2 -m

X211(α0γ)0s{1|0000} x1 x2 x3 x4 m{2y|001

212 } -x1 x2 -x3+1

2 -x4+12 m

{1’|00012 } x1 x2 x3 x4+1

2 -m{1|01

2012 } x1 x2+1

2 x3 x4+12 m

TABLE 7.5: Representative operations of the space group P211(α 12 γ)0s described in prim-

itive setting and in the X-centered one. Both generalized Seitz-type notation (left column)and symmetry cards as used in JANA2006 are listed. The labels -m or m indicate whetherthe operation includes the time inversion (-m) or not (m). The remaining operations areobtained by the internal product of the group.

The refinement from 162 independent magnetic reflections (averaged from246 reflections collected on D23 at 9 K with Rint = 2.50) confirmed the superspacesymmetry X211(α0γ)0s for AF2 phase, and therefore a spin modulation accordingto a superposition in quadrature of mG2 and mG1 modes [Table 7.3 shows that theX211(α0γ)0s results from combining mG1 and mG2 with Δφ = 1

4 , 34 ]. The refined

parameters are listed in Table 7.6 and a scheme of the magnetic structure is plottedin Fig. 7.4. The moments rotate according to an ellipse in the ub plane, where u isthe easy direction in the ac plane. In this case, in contrast with the AF3 phase, thechirality of the two Mn1 and Mn2 spin chains is the same, see Fig. 7.4 (a), giving riseto a net electrical polarization by some mechanism such as the spin-orbit driveninverse Dzyaloshinskii-Moriya effect. As it can be seen in Table 7.6 the deviationfrom a circular spiral of the modulations of both Mn atoms is very significant. Theelliptical orbits have semi-axes of [4.1(1) μB, 3.26(7) μB] and [3.6(1) μB, 2.97(6) μB],for Mn1 and Mn2 respectively. The direction u of the magnetic moments also dif-fers between the two atoms, forming an angle with the a axis of about 37(2)˝ forMn1 and of 30(2)˝ for Mn2. These differences between the two modulations arebasically due to the strong difference of the modulation amplitude along the z axis(see Table 7.6). While the x and y modulation amplitudes have similar magnitudefor both atoms, the amplitude along z differs substantially, and changes both thespiral plane and the elliptical orbit of the magnetic moments.

Table 7.6 also shows the results of a refinement where the amplitudes of themodulations along the three crystallographic directions of the two atoms were

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92 CHAPTER 7. DETAILED DESCRIPTION OF THE MAGNETIC STRUCTURES OF MNWO4

Mni Msx Ms

y Msz Mc

x Mcy Mc

z |mmax| |mmin|i = 1 0.0* 3.26(7) 0.0* 3.28(8) 0.0* 2.49(6) 4.1(1) 3.26(7)i = 2 0.0* -2.97(6) 0.0* -3.11(8) 0.0* -1.78(6) 3.6(1) 2.97(6)RF(obs)= 3.54%, RwF(obs)= 3.45%, RwF2 (obs)= 6.20%

i = 1 0.0* 3.14(3) 0.0* 3.21(4) 0.0* 2.14(5) 3.86(8) 3.14(3)i = 2 0.0* -3.14** 0.0* -3.21** 0.0* -2.14** 3.86(8) 3.14(3)RF(obs)= 3.94%, RwF(obs)= 3.75%, RwF2 (obs)= 6.71%

* Constrained by symmetry. ** Constrained manually.

TABLE 7.6: Refined parameters of the magnetic modulations (μB) of the multiferroicphase of pure MnWO4 (9 K) in the superspace group X211(α0γ)0s. The results of con-strained refinement forcing the two symmetry-independent atoms to have equal spiralmodulation is also included. Maximum and minimum values of the ordered momentsare given in the last two columns, i. e. the semi-major and semi-minor axes of the helixes.

forced to be, respectively, of equal magnitude but with opposite sign. These re-strictions make equal the elliptical helix modulations of the two atoms, except fora π phase shift, and reduces the number of refinable parameters to three. This isthe model of phase AF2 that was proposed in Ref. [87]. In this constrained re-finement, the obtained elliptical helix, forced to be common to both atoms, has3.86(5) μB and 3.14(3) μB as semi-axes and its plane forms an angle of 34(2)˝ withthe a axis, in agreement with the values reported in Ref. [87]. In this constrainedmodel the two superimposed magnetic modes are such that the mG2 mode is re-stricted to the plane xz, while the mG1 mode is limited to the y direction. In thisway, the sign correlation of neighboring spins is the same for the two modes. Thismodel neglects those degrees of freedom in the spin arrangement that, althoughallowed by symmetry, have sign correlations of neighbouring spins which are notfavored by the exchange coupling.

The simplification included in the model proposed in Ref. [87] is consistentwith the prevailing role of the isotropic exchange energy, but the better resultsof the refinement of a non-constrained model, only restricted by the superspacesymmetry of the phase, demonstrates that the modulations of the two Mn atomshave significant differences, due to some non-negligible contribution of spin-modecomponents with unfavourable exchange energies. Anisotropic effects beyond theexchange interaction, although weak, are an essential ingredient of multiferroics,and this feature of the magnetic structure of the AF2 phase, demonstrated by thepresent study, can be considered one of them. This is similar to what happens

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7.3. AF2 MULTIFERROIC PHASE 93

in weak ferromagnets, where the magnetic ordering exhibits ferromagnetic corre-lations along some directions, despite being unfavourable from the viewpoint ofexchange energy. They appear as secondary effects because they are permitted bysymmetry. The application of superspace symmetry allows to predict similar ef-fects in incommensurate magnetic phases, distinguishing the rigorous restrictionscoming from symmetry, from those that are only approximate and are associatedwith the overwhelming role of the isotropic exchange interactions.

b

a

Mn2 Mn1

(a) (b)

Mn22 Mn11

Mn2

Mn1 c

a

(a)

(b)

FIGURE 7.4: AF2 multiferroic magnetic structure of MnWO4 at 9 K. (a) Projection alongc axis showing the chirality of the structure. (b) Projection along the b axis. Only thenon-periodic modulation within a region 4a ˆ 2b ˆ 2c is represented.

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94 CHAPTER 7. DETAILED DESCRIPTION OF THE MAGNETIC STRUCTURES OF MNWO4

7.4 Atomic modulations

Apart from the constraints on the magnetic modulations, the superspace symme-try of an incommensurate magnetic phase also provides information on the possi-ble displacement modulations of the atoms positions.

Although the structural modulations induced by magnetostructural couplingare very weak (as discussed in Chapter 6), and have not been considered in themeasurements reported in this chapter, it is important to have in mind their pos-sible existence and their expected characteristics. The ferroelectric behavior ob-served in MnWO4 is in fact part of this induced structural distortion resulting fromthe magnetic symmetry break, and corresponds to a k = 0 polar structural modu-lation. In this section we present the symmetry constraints of the atomic structuresof the AF3 and AF2 magnetic phases.

The structural distortions that result from magnetostructural coupling withthe magnetic ordering are subject to the same symmetry than the actual magneticordering. This means that the atomic positions can in principle become modulated,but their modulations will be restricted by the same superspace group operationsas the magnetic modulations. Their effect is however different due to the invari-ance of the structural modulation for time reversal. Hence, the atomic modulationfunctions uν(x4) and uμ(x4) that describe the atomic displacements of the symme-try related ν and μ atoms from their basic positions rlν and rlμ are related as:

uμ(RI x4 + HR ¨ rν) = uν(x4) (7.19)

and thus, the symmetry operation t11|00012u, common to both phases, force the

possible displacive modulations of any atom to the condition:

ui(x4 +12) = ui(x4) (7.20)

This means that any possible induced structural modulation is limited to evenharmonics. Remember that in Chapter 2 the atomic modulation functions wereFourier expanded, being periodic with period 1:

Aμ(x4) = Aμ,0 +ÿ

n=1,...

[Aμ,ns sin(2πnx4) + Aμ,nc cos(2πnx4)] (7.21)

and taking into account equation (7.20) it is straight forward that the atomic mod-

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7.4. ATOMIC MODULATIONS 95

ulations are related to even harmonics.ÿm=1,...

[ui,ms sin(2πm(x4 +12)) + ui,mc cos(2πm(x4 +

12))]

=ÿ

n=1,...

[ui,ns sin(2πnx4) + ui,nc cos(2πnx4)] (7.22)

Ñ n = 2m (7.23)

This is a quite general feature of single-k incommensurate magnetic phases(see Ref. [66]), restricting possible satellite structural diffraction peaks to even nindices, while magnetic ones are necessarily odd ones3 if the modulation becomesanharmonic. Second order satellites due to induced structural modulations, al-though weak, are indeed quite common [23, 100, 157, 158], and they have beenalso found in MnWO4 by synchrotron X-ray diffraction [100].

Note that equation (7.20) implies that the non-magnetic atomic modulationsof Mn1 and Mn11 (as those of Mn2 and Mn22), related by the centering X, are identi-cal. In principle, provided that second order satellites are strong enough to be mea-surable, the atomic modulated structure could be determined separately from themagnetic one using the non-magnetic superspace group (with modulation wavevector 2k), resulting from substituting the operation t11|0001

2u by t1|00012u, and

applying then the usual procedures that have become standard for non-magneticincommensurate structures. This is however not necessary and could be mislead-ing, because magnetic superspace symmetry provides an unified framework thatconstrains all degrees of freedom, and clarifies the correlation between the con-straints imposed upon magnetic and non-magnetic degrees of freedom.

The Mn and W atoms are kept invariant by the symmetry operation t2y|0012

12u,

which is common both to AF3 and AF2 phases, and this implies, according to theanalogous of equation (7.3) that their displacive modulations must satisfy:

ui(´x4 +12) = 2y ¨ ui(x4) (7.24)

3t11|000 12 u forces the magnetic modulations to odd harmonics

Mi(x4 +12) = ´Mi(x4)ÿ

n=1,...

[Mi,ns sin(2πn(x4 +12)) + Mi,nc cos(2πn(x4 +

12))] = ´ ÿ

m=1,...

[Mi,ms sin(2πmx4) + Mi,mc cos(2πmx4)]

ÿn=1,...

[Mi,ns sin(2πn(x4 +12)) + Mi,nc cos(2πn(x4 +

12))] =

ÿm=1,...

[Mi,ms sin(2πmx4 + π)) + Mi,mc cos(2πmx4 + π)]

Ñ n = 2m + 1

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96 CHAPTER 7. DETAILED DESCRIPTION OF THE MAGNETIC STRUCTURES OF MNWO4

with i any Mn or W atom in the paramagnetic unit cell.

Hence, the x and z components of these modulations are restricted to sineterms while the y component can only contain cosine terms:

uiα(x4) = u1iα sin(4πx4) (α = x, z) (7.25)

uiy(x4) = uoiy + u1

iy cos(4πx4) (7.26)

being i = Mn, W and uoiy being related to k = 0.

The displacive modulations that can happen to W and Mn atoms by any kindof magnetostructural coupling are therefore π

2 shifted with respect to the magneticmodulation of the Mn atoms, see equation (7.9) and (7.10), independently of thespecific mechanism at play.

In the AF3 phase, any pair of atoms i = 1, 2, related by t1|01201

2u, as Mn1 andMn2, have their structural modulations related by:

u2(´x4 +12) = ´u1(x4) (7.27)

which according to equations (7.25) and (7.26) implies equal amplitudes for the x, zsine terms, and opposite ones for the y amplitudes. Mn and W are allowed to havehomogeneous (k = 0) displacements along y, see equation (7.26), but equation(7.27) implies that in the AF3 phase each pair of atoms related by inversion willhave opposite homogeneous displacements, consistently with the centrosymmet-ric point-group symmetry 2/m of this phase. Something similar happens for themore general modulations of the oxygen atoms in general positions that can havean homogeneous components along the three crystallographic directions. In con-trast, in the AF2 phase, as the inversion symmetry operation is no longer valid, theconstraint limiting the homogeneous displacements along y of inversion-relatedatoms to opposite values disappears and a polar distortion mode is allowed.

Summarizing both AF3 and AF2 phases may exhibit a magnetic inducedstructural modulation with 2k as primary modulation wave vector. In the AF3phase this structural modulation maintains an average structure with P2/c sym-metry, i. e. the same space group as in the paramagnetic phase, while the structuralmodulation in the AF2 phases looses some of its constraints and has only as sym-metry for the average structure the space group P2.

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7.5. SUMMARY AND CONCLUSIONS 97

7.5 Summary and conclusions

The two incommensurate magnetic phases of the multiferroic MnWO4 have beenre-analyzed using the superspace formalism. The superspace group correspond-ing to each phase has been identified, and the structures have been re-determinedunder these symmetries, using single-crystal non-polarized neutron-diffractiondata. The spin modulations of the Mn atoms have been fully determined, distin-guishing those features forced by symmetry and fulfilled exactly, from those onlyapproximate and essentially caused by the strong weight of the isotropic exchangeinteraction. The study shows that in the antiferromagnetic phase AF3, the modu-lations of the two Mn atoms in the unit cell can have a small cycloidal component,exactly equal but with opposite chiralities in the two atoms, such that their effectcancel and they cannot induce any electric polarization. In contrast, in the AF2phase, the addition of a second magnetic mode to the spin modulations breaksthe symmetry relation between the two manganese atoms, the two modulationshave then chiralities of the same sign, and their effect add up giving rise an electricmacroscopic polarization.

The superspace group of each phase also determines the magnetic point-group governing its tensor properties. The ferroic properties of the multiferroicAF2 phase can then be characterized by the point-group symmetry break2/m11 Ñ 211, i. e. by the ferroic species 2/m11F211 [159]. This ferroic species im-plies that there can only be two types of ferroic domains in the AF2 phase, relatedby the space inversion lost in the symmetry break or equivalently by the mirrorplane. Thus, the correlated switching properties of electric polarization and mag-netic modulations in phase AF2 can be readily predicted: space inversion switchespolarization, while it keeps invariant the amplitude of the mG2 irrep mode andchanges sign of the amplitude of the mG1 irrep modulation. This means that thetwo domains with opposite polarization have also spin modulations of oppositechirality related by inversion.

The application of superspace formalism to the analysis of incommensuratemagnetic phases is a promising new development with yet few examples in theliterature. We explain here its application on MnWO4 and demonstrate how apowerful it is to describe and use systematically the symmetry properties of in-commensurate magnetic structures.

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98 CHAPTER 7. DETAILED DESCRIPTION OF THE MAGNETIC STRUCTURES OF MNWO4

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Chapter 8Magnetic structure determination

and ferroelectric properties ofMn1´xCoxWO4 (x = 0.05, 0.10, 0.15

and 0.20)

Chapter 8 is devoted to the determination of the magnetic structures, thestudy of the ferroelectricity and the inter-relation between the magneticand electric properties of the Mn1´xCoxWO4 series at zero field. For that

purpose we have used bulk magnetic measurements, neutron scattering and py-roelectric measurements on single crystals. The results for each composition arepresented first, followed by a summary of the results, our contribution to the mag-netic phase-diagram and a preliminary electric phase-diagram.

By the time the work on this thesis began, there was a single publication bySong and co-workers [125] discussing the influence of cobalt substitution for man-ganese in MnWO4. However, since the proposed magnetic structures were basedon neutron-powder-diffraction data, the refined structures were not unambigu-ously determined and important details of them were not provided (as alreadymentioned in Chapter 4, neutron powder-diffraction sometimes is not enough forthe determination of magnetic structures). That is why, in the conclusions of thisfirst paper on the Co substitution the authors called for "further work using sin-gle crystal samples to resolve the uncertainties" of the analyses of their powder-diffraction data.

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100CHAPTER 8. MAGNETIC STRUCTURE DETERMINATION AND FERROELECTRIC PROPERTIES OF

MN1´XCOXWO4 (X = 0.05, 0.10, 0.15 AND 0.20)

During this years, new important features of the magnetic phase diagramemerged from the work by other groups [126–131] which were published in par-allel with the results obtained during the work on this thesis. At the end of thischapter there is a section devoted to the comparison and discussion about the re-sults obtained by other groups and ours.

8.1 Mn0.95Co0.05WO4

8.1.1 Bulk magnetic response

The temperature variation of the susceptibilities along the b axis and along themagnetically easy and hard axes (confined within the ac plane as determined be-low) are presented in Fig. 8.1(a). The easy axis is called α, whereas the hard axisis called α + 90˝. For all orientations the susceptibilities have the highest value atTN = 13.2(2) K. At lower temperature, there is an anomaly in the susceptibilityalong b at TN2 = 12.2(2) K which can be attributed to a magnetic transition. Noevidences of this transition are observed in the susceptibilities along α and α+ 90˝,within experimental resolution. The magnetic anisotropy is confirmed by measur-ing the angular dependence of the magnetic susceptibility by rotating the crystalaround the b axis, see Fig. 8.1(b). According to these measurements, the minimumof the susceptibility, which indicates the easy axis, occurs at 15(1)˝ from a, and thehard axis, α + 90˝, is located approximately 107(1)˝ from a.

The difference of susceptibilities along the different directions at high tem-peratures (in the paramagnetic region) is found to be zero within experimentalresolution. Figure 8.1(c) shows the inverse of the magnetic susceptibilities alongthe crystallographic axes, a, b and c: they follow the Curie-Weiss law with an effec-tive magnetic moment μe f f = 5.88(8) μB, which is in a very good agreement withthe expected contribution from the Mn2+ and the effective spin only moment ofthe Co2+ ions (5.83 μB). The paramagnetic Curie temperature ΘP = -64(3) K ispractically the same for all the directions. The latter, together with the negligibleorbital contribution is in agreement with the lack of anisotropy in the paramagneticregion [43–45].

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8.1. MN0.95CO0.05WO4 101

0 100 200 300 4000

1

2

3

χ-1 (1

04 CG

S)

Temperature (K)

bc

a

TN

P

H = 1 kOe

0 45 90 135 180

1.3

1.4

1.5

1.6H = 1 kOe

16 K 5 K

c

χ (1

0-4 C

GS

)

Angle (o)

a+90o-a

(c)

2 4 6 8 10 12 14 16 181.3

1.4

1.5

1.6 H = 1 kOe

TN2

Temperature (K)

χ (1

0-4 C

GS

)

b

+ 90o TN

12 13

1.55

1.60

TN2 T

N

(b)

(a)

FIGURE 8.1: (a) Temperature dependence of dc susceptibility of Mn0.95Co0.05WO4 alongthe b axis and along two distinctive directions within the ac plane, namely at an angleα of about 15˝ with respect to the a axis and perpendicular to it α + 90˝. (b) Angulardependencies of susceptibility within the ac plane (measurement done in collaborationwith the group leaded by Prof. A. A. Mukhin). (c) Inverse of the susceptibility alongthe a, b and c (no difference was found also between α and α + 90˝ in the paramagneticregion, data not shown).

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102CHAPTER 8. MAGNETIC STRUCTURE DETERMINATION AND FERROELECTRIC PROPERTIES OF

MN1´XCOXWO4 (X = 0.05, 0.10, 0.15 AND 0.20)

c

a

b

a

b

a

b

a

(a)

(b) (c)

(d)

mb mbmαmα

T = 6 K T = 2 K

0 1 20

1

2

RF= 20.3%RF2w = 16.4%

T = 12.6 KCollinear model

(Fca

lc)2 (a

rb. u

nits

)

(Fobs)2 (arb. units)

0.0 0.5 1.0 1.50.0

0.5

1.0

1.5

RF = 5.34%

RF2w = 6.70%

T = 2 KAF2 phase

(Fca

lc)2 (a

rb. u

nits

)

(Fobs)2 (arb. units)

0.0 0.5 1.00.0

0.5

1.0

RF = 8.44%

RF2w = 9.23%

T = 6 KAF2 phase

(Fca

lc)2 (a

rb. u

nits

)

(Fobs)2 (arb. units)

0 1 20

1

2

RF = 20.4%

RF2w = 14.5%

T = 12.6 KCycloidal modelAF3 symmetry

(Fca

lc)2 (a

rb. u

nits

)

(Fobs)2 (arb. units)

0 1 20

1

2

RF= 14.8%RF2w = 10.5%

T = 12.6 KCycloidal modelAF2 symmetry

(Fca

lc)2 (a

rb. u

nits

)(Fobs)

2 (arb. units)

FIGURE 8.2: Magnetic models for Mn0.95Co0.05WO4 for data collected at 12.6 K:(a) model 1 (sinusoidal AF3); (b) model 2 (cycloidal AF2), with opposite chirality for Mn1

and Mn2 and (c) model 3 (cycloidal AF2), with equal chirality for Mn1 and Mn2. (d)Magnetic model for data collected at 6 K and 2 K. The yellow arrows sign the chiralityof the spins. All the models are accompanied by their corresponding agreement plots. Aunique model is given for the data at 6 K and 2 K, the ellipse that envelops the amplitudesis drawn above the agreement plots.

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8.1. MN0.95CO0.05WO4 103

8.1.2 Neutron diffraction

The magnetic structures of Mn0.95Co0.05WO4 were studied on two instruments:6T2 in the Laboratoire Léon Brillouin (LLB) at Saclay and D23 in the Institut LaueLangevin (ILL) in Grenoble (in both cases the sample was mounted inside a mag-net). The propagation vector that describes the periodicity of the magnetic phaseswas refined from centering about thirty magnetic reflections at 2 K and resulted ink = [-0.216(1), 1

2 , 0.461(1)].

The temperature evolution of several magnetic reflections appearing belowTN = 13.2 K did not show any anomaly which could be related to a magnetic phasetransition at „ 12.2 K, expected from the bulk magnetic data. Since the temperaturerange for the high temperature magnetic phase is only 1 K wide (12.2 K - 13.2 K)and furthermore, it is close to the Néel temperature (weak magnetic reflections),collecting data at the appropriate temperature and refining those data is a ratherdifficult task. We collected data at 12.6 K (this refers to the thermometer usedin the neutron experiment, and might be few tents of degree different from thetemperature read in the magnetic measurements). Three following models havebeen tested to explain the collected data, and their details are given in Table 8.1:

(i) Model 1 (Sinusoidal AF3): It is a collinear structure with the moments in theac plane at about 11(2)˝ from the a axis and sinusoidally modulated. Thisstructure is similar to the AF3 phase of the pure compound and agrees withthe model proposed by Song and co-workers in Ref. [126]. The structure isplotted in Fig. 8.2(a).

(ii) Model 2 (Cycloidal AF3): It corresponds to a cycloidal structure with oppo-site chirality for Mn1 and Mn2. The moments rotate in the ub plane, whereu is a direction in the ac plane 15(2)˝ from a. The chirality of the Mn1 andMn2 atoms is opposite which maintains the symmetry of the AF3 phase asexplained in Chapter 7. Figure 8.2(b) depicts this structure. This structure isan analogue of the alternative antiferroelectric AF3 structure of MnWO4, assuggested by the superspace analysis provided in Chapter 7.

(iii) Model 3 (Cycloidal AF2): This model is similar to the one of the multiferroicphase of the pure compound. The differences with respect to that structureare related to the amplitudes of the moments and the inclination of the rota-tion plane within the ac plane. Thus, the structure is cycloidal, with the spinsrotating in the ub plane, u being within the ac plane 15(2)˝ from a. Both man-ganese atoms in the crystallographic unit cell have the same chirality, and

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104CHAPTER 8. MAGNETIC STRUCTURE DETERMINATION AND FERROELECTRIC PROPERTIES OF

MN1´XCOXWO4 (X = 0.05, 0.10, 0.15 AND 0.20)

therefore, they break the spacial inversion. The spin arrangement is plottedin Fig. 8.2(c).

The magnetic sinusoidal AF3 (model 1) and cycloidal AF3 (model 2) spin ar-rangements have the same symmetry as the AF3 phase of the pure compound andtherefore they take its name (even if the model 2 has more freedom than the model1, in the sense that more parameters are refined to describe the structure). Thesame reasoning is applied to model 3, which has the symmetry of the so-calledAF2 phase. Although among those plausible models the AF2 fits the data slightlybetter, its validity should be taken with some precautions since the residual fac-tors of all the proposed models are not very good. This may be due to the fact theexperiment was done close to the Néel temperature and inside a magnet, whichintroduces noise to the data. The magnetic models are accompanied by their cor-responding agreement plots. The compatibility of the models with polarizationresults should be compared to try to choose one model among them.

Temperature T = 12.6 K T = 12.6 K T = 12.6 K T = 6 K T = 2 KMagnetic phase (i) AF3 (ii) AF3 (iii) AF2 AF2 AF2Instrument D23 D23 D23 6T2 6T2

k [-0.216(1), 0.5, 0.461(1)]

Mn1 �(m)(μB) 2.72(3) 2.63(3) 2.38(4) 4.26(6) 4.67(4)φu 0˝ 0˝ 0˝ 0˝ 0˝θu 79(2)˝ 75(2)˝ 80(2)˝ 73(2)˝ 77(2)˝�(m)(μB) 0.78(9) 1.24(6) -3.51(6) -4.20(5)φv 90˝ 90˝ 90˝ 90˝θv 90˝ 90˝ 90˝ 90˝

Mn2 ux,z2 = ´ux,z

1 ux,z2 = ´ux,z

1 ux,z2 = ´ux,z

1 ux,z2 = ´ux,z

1 ux,z2 = ´ux,z

1v2 = v1 = 0 vy

2 = vy1 vy

2 = ´vy1 vy

2 = ´vy1 vy

2 = ´vy1

RΔϕ kz2

kz2

kz2

kz2

kz2

RF2/% 20.3 20.4 14.8 8.44 5.26RF2w/% 16.4 14.5 10.5 9.23 6.58RF/% 26.9 26.4 22.0 5.46 3.65χ2 12.8 10.2 5.34 9.79 7.03

TABLE 8.1: Parameters that describe the magnetic structures of Mn0.95Co0.05WO4. Mag-netic atoms Mn1 and Mn2 are located in the crystallographic unit cell at [ 1

2 , 0.6839(7), 14 ]

and [ 12 , 0.3161(7), 3

4 ], respectively. Three models are given to describe the magnetic struc-ture at 12.6 K: (i) model 1 (sinusoidal AF3), (ii) model 2 (cycloidal AF3) and (iii) model 3(cycloidal AF2).

Regarding the low temperature magnetic phase, the refinements were donefrom the integrated intensities of about a hundred reflections at 6 K and 2 K, re-spectively. The details of the structure are given in Table 8.1. The low temperature

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8.1. MN0.95CO0.05WO4 105

phase corresponds to a cycloidal structure that has the same symmetry as the AF2phase of the pure MnWO4, so, accordingly this phase is named AF2. The momentsare rotating in the ub plane and the amplitudes are enveloped by an ellipse. Theorientation of the direction u changes a few degrees from 73(2)˝ to 77(2)˝ whentemperature decreases from 6 K to 2 K. The main difference concerns their ampli-tudes of the ordered magnetic moments; the lower the temperature is, the biggerthe amplitudes are. It is worth to pay attention on the way the amplitudes increaseand how the eccentricity of the ellipse evolves. If the eccentricity is defined as

ε =

gffe1 ´ �(m)2

�(m)2, (8.1)

then, it changes from ε(6 K) = 0.54(3) to ε(2 K) = 0.44(3). The development of themagnetic component along the b axis, makes the ellipse more circular as temper-ature decreases and therefore, the magnetic anisotropy within the ub plane de-creases. Figure 8.2(d) shows the magnetic structures model for the multiferroicphase and a plot of the ellipses that envelop the amplitudes of the magnetic mo-ments at T = 6 K and T = 2 K, respectively.

8.1.3 Electric polarization

2 4 6 8 10 12 140

10

20

30

40

Pb (μC

/m2 )

Temperature (K)

TFEAF2 (Pb)

P = 0

FIGURE 8.3: Temperature dependence of the spontaneous electric polarization along theb axis.

Spontaneous electric polarization appears only along the b axis belowTFE = 12.0(1) K and increases up to about 37 μC/m2 at 2 K as shown in Fig. 8.3.

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106CHAPTER 8. MAGNETIC STRUCTURE DETERMINATION AND FERROELECTRIC PROPERTIES OF

MN1´XCOXWO4 (X = 0.05, 0.10, 0.15 AND 0.20)

The temperature at which the polarization arises, TFE, coincides with the tempera-ture where the anomaly in the magnetic susceptibility along the b axis is observed,TN2, see Fig. 8.1(a), that is, when entering the AF2 phase.

The fact that the polarization arises below 12.0(1) K can help us to discard thepolar Model 3 to describe the magnetic structure at 12.6 K. So, we are left with thesinusoidal collinear structure and the cycloidal structure with opposite chirality forMn1 and Mn2. Anyway, as mentioned before, the data refinement is not conclusive.

The polarization along the b axis could be indeed predicted from the sym-metry of the AF2 magnetic structure [X211(α0γ)0ss], as it is the case for the pureMnWO4 (see Chapter 7).

8.2 Mn0.90Co0.10WO4

8.2.1 Bulk magnetic response

The temperature dependence of the dc susceptibility with field applied along theprincipal crystallographic directions is given in Fig. 8.4(a). A clear maximum,which could be associated to the Néel temperature, is seen at TN = 13.2(1) K for allthe three directions. It is worth noting that the a- and c-axis susceptibilities havea similar value at TN , which is bigger than the one for the b axis. With furtherdecrease of the temperature another small change in the susceptibility slope takesplace at TN2 = 11.4(1) K, more visible along the c axis, signaling a second magnetictransition. The two characteristic temperatures are in a good agreement with thedata reported in Ref. [126] and are associated with the appearance of the AF3phase below TN and with the so-called AF2’ phase below TN2. Though below TN

the a-axis susceptibility decreases faster than the one along the c axis, they bothchange in a similar fashion and in disparity with the relatively small decrease ofthe b-axis susceptibility.

Above TN , the a- and c-axis susceptibilities are very close and higher thanthe b-axis susceptibility. The observed anisotropy is rather different from the onereported in Ref. [126]. However, this discrepancy and the magnitude of the mea-sured anisotropy itself should be taken with some precautions until careful mea-surements on a crystal with similar dimensions along the three directions are re-ported, or after proper corrections for sample shape and size are made [160]. Attemperatures above 30 K, the susceptibilities along all the three directions fol-

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8.2. MN0.90CO0.10WO4 107

-100 0 100 200 300 4000

1

2

3

P

TN

abc

χ-1 (1

04 CG

S)

Temperature (K)

H = 1 kOe

0 2 4 6 8 10 12 14 16 18 201.2

1.4

1.6TN

TN2

Temperature (K)

a

c

χ (1

0-4 C

GS

) b

H = 1 kOe

(b)

10 12 141.45

1.50

1.55c axis

TN2

TN

(a)

FIGURE 8.4: (a) Temperature dependence of dc susceptibilities at field of 1 kOe appliedalong the principal crystallographic directions. (b) Temperature dependence of the in-verse susceptibilities along the same directions. TN and θP stand for Néel and asymptoticparamagnetic temperatures respectively, while TN2 refers to the small change in the sus-ceptibility slope, tentatively ascribed to a second transition.

low the Curie-Weiss law, see Fig. 8.4(b), with an effective magnetic moment ofμe f f = 5.74(1) μB, which is in a very good agreement with the expected contribu-tion from the Mn2+ and the effective spin only moment of the Co2+ ions(μe f f = 5.75 μB). The paramagnetic Curie temperature ΘP = -71(3) K is practicallythe same for all the crystallographic directions.

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108CHAPTER 8. MAGNETIC STRUCTURE DETERMINATION AND FERROELECTRIC PROPERTIES OF

MN1´XCOXWO4 (X = 0.05, 0.10, 0.15 AND 0.20)

8.2.2 Neutron diffraction

The two magnetically ordered states in Mn0.90Co0.10WO4, evidenced by the sus-ceptibility measurements, are reflected in the variation of the integrated intensitiesof two representative magnetic reflections shown in Fig. 8.5(a). Below TN „ 13 Kit orders in the so-called AF3 phase. Then at TN2 „ 11 K, a second transition oc-curs towards the so-called AF2’ phase [125, 126], as revealed by the accident in thecurves of Fig. 8.5(a) and in agreement with magnetic susceptibility measurements.

2 4 6 8 10 12 14 160.0

0.5

1.0

1.5

2.0

Temperature (K)

(0.5 0 0) AF4

(Fob

s)2 (arb

. uni

ts)

2 4 6 8 10 12 14 16

500

1000

1500

Temperature (K)

(0 0 0)+ k

(0 0 1)- kP

aram

agne

tic

AF3

(Fob

s)2 (arb

. uni

ts)

AF2’

TN2 TN

(b)

(a)

FIGURE 8.5: Integrated intensities of magnetic reflections in zero applied field mon-itored as the temperature increases. (a) Magnetic satellites that correspond tok = [-0.222(1), 1

2 , 0.472(1)]. The error bars are inside the symbols. Vertical lines indicatethe two magnetic transitions. (b) The ( 1

200) reflection characteristic of the AF4 commen-surate phase.

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8.2. MN0.90CO0.10WO4 109

Temperature T = 12 K T = 12 K T = 9 K T = 2 KMagnetic phase AF3 AF3 AF2’ AF2’

k [-0.222(1), 12 , 0.472(1)]

Mn1 �(m)(μB) 2.53(7) 2.55(7) 3.71(1) 4.45(2)φu 0˝ 0˝ 0˝ 0˝θu 124.7(5)˝ 123.3(5)˝ 115.5(8)˝ 99(2)˝

�(m)(μB) -0.13(2) 2.80(1) 4.03(2)φv 90˝ 0˝ 0˝θv 90˝ 25.5(8)˝ 9(2)˝

Mn2 ux,z2 = - ux,z

1 ux,z2 = - ux,z

1 ux,z2 = - ux,z

1 ux,z2 = - ux,z

1vy

2 = vy1 = 0 vy

2 = vy1 vx,z

2 = - vx,z1 vx,z

2 = - vx,z1

Δϕ kz2

kz2

kz2

kz2

RF2/% 15.7 15.3 6.51 5.67RF2w/% 13.6 12.6 6.94 6.16RF/% 17.6 17.3 8.49 6.65χ2 5.53 4.82 7.00 7.14

TABLE 8.2: Parameters that describe the magnetic structures of Mn0.90Co0.10WO4.Mn1 and Mn2 are located in the crystallographic unit cell at [ 1

2 , 0.6847(2), 14 ] and

[ 12 , 0.3153(2), 3

4 ], respectively.

After centering 20 magnetic reflections and refining the propagation vectorat 12 K, 9 K and 2 K (k = [-0.222(1), 1

2 , 0.472(1)]), which within error bars is temper-ature independent, 680 magnetic peaks were collected at each temperature. Thedetails of the refined magnetic structures are given in Table 8.2. The spin arrange-ments are depicted in Fig. 8.6 together with the observed and calculated squaredstructure factors. As for the pure and 5%-Co-doped compositions, two models canfit the data in the AF3 phase: a collinear structure with sinusoidally modulatedamplitudes or a cycloidal order with different chirality for Mn1 and Mn2. In thecollinear structure the moments lay in the ac plane, along a direction u making anangle ψ = 34.7(5)˝ (θu - 90˝) with the a axis, with maximum amplitude of 2.53(3) μB.They are confined in the equatorial plane of the MO6 octahedra and perpendicularto the b axis, see Fig. 8.6(a). In the cycloidal model the moments have an ad-ditional component along the b axis of ´0.13(2) μB, see Fig. 8.6(b). Even if thelatter structure is a cycloidal one, both Mn atoms in the crystallographic unit cellhave opposite chirality and hence, the structure is non-polar. Nevertheless, be-low 10.8 K another component (v) appears perpendicular to the existing one (u).Both u and v, are in the ac plane. As a consequence, the structure becomes anelliptical helix in this ac plane, see Fig. 8.6(c), the so-called AF2’ phase [a brief su-perspace analysis is provided in Appendix E, where it is proved that the symmetryof the structure corresponds to Xm11(α0γ)ss]. From the analysis of the magnetic

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110CHAPTER 8. MAGNETIC STRUCTURE DETERMINATION AND FERROELECTRIC PROPERTIES OF

MN1´XCOXWO4 (X = 0.05, 0.10, 0.15 AND 0.20)

c

a

b

a

(a)

(b) (c)

mv mv

mumu

T = 9 K T = 2 K

c

a

0.0 2.5 5.00.0

2.5

5.0

(Fobs)2 (arb. units)

(Fca

lc)2 (a

rb. u

nits

)

RF = 17.6%RF2w = 13.6%

T = 12 KCollinear AF3

0.0 2.5 5.00.0

2.5

5.0

RF = 17.3%RF2w = 12.6%

T = 12 KCycloidal AF3

(Fca

lc)2 (a

rb. u

nits

)

(Fobs)2 (arb. units)

0.0 0.8 1.60.0

0.8

1.6

RF = 6.65 %RF2w = 6.94 %

(Fca

lc)2 (a

rb. u

nits

)

(Fobs)2 (arb. units)

T = 9 KAF2’

0.0 12.5 25.00.0

12.5

25.0

(F ca

lc)2 (a

rb.u

nits

)

RF = 8.49%RF2w = 6.16%

T = 2 KAF2’

(Fobs)2 (arb.units)

FIGURE 8.6: Magnetic structure models of Mn0.90Co0.10WO4 in zero field with their corre-sponding agreement plots. (a) the collinear and (b) cycloidal models for the AF3 magneticphase at 12 K and (c) the cycloidal model for the multiferroic AF2’ phase. The ellipses thatenvelop the amplitudes of the moments in the multiferroic phase are plotted at 9 K and2 K. The yellow arrow mark the chirality of the moments. The corresponding magneticmodels are described in the text and Table 8.2.

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8.2. MN0.90CO0.10WO4 111

intensities we can appreciate a change in the orientation of the principal axes ofthe ellipse within the ac plane when decreasing the temperature. The angle thatthe semi-major axis of the ellipse makes with the c˚ axis decreases from 124.8(5)˝down to 99(2)˝ on cooling from 12 K to 2 K. Moreover, in the AF2’ multiferroicphase there is an apparent evolution of the eccentricity of the ellipse, defined as inequation (8.1), with temperature. From 9 K to 2 K the eccentricity changes fromε = 0.66(1) to 0.42(1). The fast development of the magnetic component perpendic-ular to u as the temperature decreases in the AF2’ phase, results in a more circularcycloid. Figure 8.6(c) shows the shape and orientation of the ellipses that envelopthe magnetic moments.

We did not observe any sign of the collinear AF4 phase with propagationvector k = (1

2 , 0, 0), see Fig. 8.5(b). Traces of this phase were reported by Songet. al. [125] for powder samples of Mn0.90Co0.10WO4. Its possible presence in oursample was carefully examined. Our conclusion is that this commensurate phasecharacterized by k = (1

2 , 0, 0) requires higher doping and does not appear in the10 % Co-doped crystal. The presence AF4 in powder samples may be due to someinhomogeneities in the composition.

8.2.3 Electric polarization

0 5 10 15 200

25

50

75

100

Pb

Pc

Pa

P (μ

C/m

2 )

Temperature (K)

5 10 151.00

1.01

1.02

1.03

εcx10

εa

Temperature (K)

ε(T

)/ε(5

K)

FIGURE 8.7: Temperature dependence of the spontaneous electric polarization along theprincipal crystallographic directions which appeared below TN2, in the AF2’ cycloidalphase. Insert shows the anomalies of the dielectric constants along a and c axes at TN2

(measured at 16 kHz).

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112CHAPTER 8. MAGNETIC STRUCTURE DETERMINATION AND FERROELECTRIC PROPERTIES OF

MN1´XCOXWO4 (X = 0.05, 0.10, 0.15 AND 0.20)

The temperature dependence of the electric polarization along a, b and c axes inzero magnetic field is shown in Fig. 8.7. Below TN2 „ 11 K electric polarizationis observed along both a and c axes, which increase up to Pa „ 100 μC/m2 andPc „ 30 μC/m2 at low temperatures. The polarization along the b axis does notexceed 3 μC/m2 and is probably due to a small misalignment from the b axis. In-sert in Fig. 8.7 shows the temperature variation of the dielectric constant alonga and c axes, which reveals peaks at TN2 where the spontaneous polarization ap-pears; the peak’s amplitude along a axis is significantly higher than along c axis.These results unambiguously show that spins in the cycloidal ferroelectric phaseAF2’ are aligned in the ac plane in agreement with our results from neutron diffrac-tion and with Refs. [125, 126]. However we observe polarization not only alongthe a axis, as expected for the predominated Mn-Mn exchange in zigzag chainsalong the c axis, but also a noticeable Pc component, the value of which is beyondpossible misalignment in the measurement direction and thus, suggests a contri-bution from the exchange along the a axis. This component of the polarization(„ 20 μC/m2) was also observed in Ref. [126]. This contribution from the exchangealong the a axis is supported by the spin-wave excitations studies in MnWO4 [118]that confirmed that the exchange coupling between chains along the a axis is asstrong as along the c axis and similarly strongly frustrated.

8.3 Mn0.85Co0.15WO4

8.3.1 Bulk magnetic response

The temperature dependence of the magnetic susceptibilities along b and alongtwo particular directions are plotted in Fig. 8.8. The latter two directions are lo-cated within the ac plane, one at about 35˝ from the -c axis (named α) and the otherperpendicular to it (called α + 90˝). The angular dependence of the susceptibilityat room temperature presented in Fig. 8.9(a), exhibits its maximum value alongα, and a minimum at α + 90˝. On the contrary, when the crystal is magneticallyordered, the maximum and minimum values are switched.

The α-axis susceptibility shows a clear maximum at 16.7(2) K, whereas alongα + 90˝ and b axes there is a less visible accident that slightly changes the slopeof the curve. This singularity could be attributed to the Néel temperature, TN .Several anomalies occur below TN which are evident for all the orientations, andsuggest changes in the magnetic order. The first occurs at T2 = 10.4(2) K, temper-

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8.3. MN0.85CO0.15WO4 113

0 5 10 15 20 25

1.2

1.3

1.4

1.5

1.6

1.7 T5 T4 T3 T2

χ (1

0-4 C

GS

)

Temperature (K)

TN

H = 1 kOe

b

+90o

FIGURE 8.8: (a) Temperature dependencies of magnetic susceptibilities ofMn0.85Co0.15WO4 along b, α (easy-axis) and α + 90˝ (hard-axis) directions. Full circles (‚)are recorded during heating of the sample and empty circles (˝) are recorded while thesample was cooled down. Dotted lines mark the anomalies of the susceptibilities duringheating of the sample.

ature at which the α + 90˝- and b-axis susceptibilities have their maximum value.However, at about 9.4(2) K, χb recovers its value and χα+90˝ increases a bit. At thispoint α-axis susceptibility significantly decreases until they reach a linear behavior.The next anomaly happens at T4 = 7.0(2) K where the slope of all the curves changesmoothly during cooling (˝), while on the contrary, with increasing the tempera-ture (‚) the change occurs in a step-like fashion along α and α + 90˝ axes. Finally,there is a small change in the susceptibility at T5 = 3.5(2) K which marks anothertransition. All the low temperature transitions are highly hysteretic, depending onwhether the temperature increases or decreases, thus implying a first-order phase-transition and phase coexistence.

A magnetic anisotropy in the paramagnetic region is clearly observed inFig. 8.9(b), where the inverse susceptibilities along b, α and α + 90˝ are plotted.The χ´1 become parallel straight lines at high temperatures and follow the Curie-Weiss law. The asymptotic paramagnetic temperatures, ΘP, obtained for the threedirections differ considerably: ΘP = -74(2) K along the b axis, ΘP = -56(2) K for theα orientation, and ΘP = -80(2) K for the orientation perpendicular to α within theac plane. Furthermore, the effective magnetic moment was found to be 5.88(5) μB,

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114CHAPTER 8. MAGNETIC STRUCTURE DETERMINATION AND FERROELECTRIC PROPERTIES OF

MN1´XCOXWO4 (X = 0.05, 0.10, 0.15 AND 0.20)

0 45 90 135 180

1.2

1.4

1.6

20 K 7.5 K 4 K

~ +90o

~

H = 1 kOeχ

(10-4

CG

S)

Angle (o)

0.38

0.40 300 K a

-c

-a

-100 0 100 200 300 4000

1

2

3H = 1 kOe

+ 90o

χ-1 (1

04 CG

S)

Temperature (K)

b

(a)

(b)

FIGURE 8.9: (a) Angular dependence of χKb at 300 K, 20 K, 7.5 K and 4 K measuredin collaboration with Prof. A. A. Mukhin. (b) Inverse of χ along b, α and α + 90˝ ofMn0.85Co0.15WO4.

higher than the one expected from the Mn2+ ions and the spin only contributionfrom the Co2+ ions (5.66 μB) and therefore suggests that the Co orbital moment is

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8.3. MN0.85CO0.15WO4 115

not completely frozen [43–45]. The anisotropy in ΘP and the fact that the effectivemagnetic moments suggest orbital contribution from the cobalt, fully support themagnetic anisotropy in the paramagnetic region of the 15%-Co-doped composi-tion.

8.3.2 Neutron diffraction

Magnetic reflections were recorded as a function of the temperature in order tofollow magnetic transitions and to determine the propagation vectors present ineach phase. In this case, magnetic reflections were monitored by performing scansalong specific directions in the reciprocal space instead of integrated intensities.

-0.26 -0.24 -0.22

4

6

8

10

12

14

16

18

AF2

AF5

(Qh, -1.5, -2Qh) T

empe

ratu

re (K

)

Qh

10.00

875.0

1700(b)

AF1

0.40 0.45 0.50 0.55 0.60

4

6

8

10

12

14

16

18

AF4

Tem

pera

ture

(K)

Qh

10.00

1250

2500

3480(a) (Qh, 1, 1)

FIGURE 8.10: Scans of the magnetic reflections as a function of the temperature.(a) Qh scan centered on the (0.5 1 1) reflection. (b) Scan along (Qh, -1.5, -2Qh) directionin the reciprocal space that shows reflections that correspond to three incommensuratepropagation vectors. Black dashed lines are guides to observe the magnetic transitions.

Figure 8.10(a) shows Qh scan of the (0.5 1 1) magnetic reflection characteris-tic of the AF4 phase as a function of the temperature, whereas, Fig. 8.10(b) showsthe temperature variation of the scans along (Qh, -1.5, -2Ql) direction in the re-ciprocal space. Along this latter direction reflections characteristic of AF1, AF2and AF5 phases are visible. Four propagation vectors are thus observedbelow TN : kAF1 = ( 1

4 , 12 , 1

2 ); kAF2 = [-0.22(1), 12 , 0.45(1)]; kAF4 = (- 1

2 , 0, 0) andkAF5 = [-0.23(1), 1

2 , 0.48(1)]. The values of the incommensurate propagation vec-tors were refined from the scans along Qh and Ql directions. The lower accuracywith respect to the other compounds comes from the coexistence of the reflectionsand therefore, the difficulty to center them due to their overlapping.

As it can be observed in Fig. 8.10(a) the magnetic phase AF4 exists below the

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116CHAPTER 8. MAGNETIC STRUCTURE DETERMINATION AND FERROELECTRIC PROPERTIES OF

MN1´XCOXWO4 (X = 0.05, 0.10, 0.15 AND 0.20)

Néel temperature down to the lowest investigated temperature, 2 K. That mag-netic peak appears at TN , then its intensity increases up to about 9 K, where itdrops down to a third of its value to become almost constant down to 2 K. In Fig.8.10(b) one can observe that the commensurate AF1 phase exists in the 9.5 - 4 Ktemperature range, while the intensity of the AF5 peak emerges below 4.5 K andgrows down to 2 K. The incommensurate peak that corresponds to the AF2 mul-tiferroic phase exhibits the most complex behavior. It shows up at about 10.5 K,slightly before the AF1 phase, and when the latter appears the intensity of the AF2reflection decreases, but it does not vanish. This AF2 phase recovers some inten-sity when the AF1 peak starts loosing it, as if it was replacing it. On the otherhand, at about 4.5 K AF5 emerges with a subsequent disappearance and decreaseof the AF1- and AF2-peak’s intensity, respectively. Taking into account this infor-mation we can associate the characteristic temperatures obtained from magneticsusceptibility data with the magnetic phases:

˛ T2 = 10.4(2) K: Emergence of AF2.

˛ T3 = 9.4(2) K: Appearance of AF1 phase and modification of AF2 (eventuallydisappearance).

˛ T4 = 7.0(2) K: Change in the AF1 phase and reappearance of AF2.

˛ T5 = 3.5(2) K: There is not any clear evidence of any change in the evolutionof the studied magnetic peaks.

Note that even if the AF5 peak vanishes abruptly at about 4.5 K [see Fig. 8.10(b)]there is not any trace of this transition in the evolution of the magnetic susceptibil-ity, and this suggests that the AF5 phase could be somehow similar to the under-lying AF2.

Intensity maps in the reciprocal space at several temperatures are depictedin Fig. 8.11, where the overlap of the reflections is very clear at 4.6 K and 1.6 K.Maps on Figs. 8.11(a), 8.11(b), 8.11(c), 8.11(d) and 8.11(e) were recorded consecu-tively and partially unveil the magnetic behavior that depend on whether the sam-ple is cooled or heated, as seen in the susceptibility data. The comparison betweenFigs. 8.11(c) and 8.11(e) demonstrate that AF5- and AF1-peak’s intensities are dif-ferent at the same temperature (4.6 K in this case) depending whether the sample isfurther cooled down between both measurements or not. The magnetic hysteresiscould be related to the martensitic nature of the transition.

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8.3. MN0.85CO0.15WO4 117

0 3 6 9

1.2

1.3

1.4

1.5

0.4 0.5 0.6

-0.28

-0.24

-0.20

3

AF2 AF5

AF1

Qh

Ql

(c)

0.4 0.5 0.6

-0.28

-0.24

-0.20

2

AF2

AF1

Qh

Ql

(b)

0.4 0.5 0.6

-0.28

-0.24

-0.20

AF2

AF1

Qh

Ql

5.000

504.0

1029

1550

1

(a)

0.4 0.5 0.6

-0.28

-0.24

-0.20(e)

5

AF2 AF5

AF1

Qh

Ql0.4 0.5 0.6

-0.28

-0.24

-0.20

4

(d)

Q

h

Ql

AF2 AF5

AF1

5

4 3 2 1

χ

(10-4

CG

S)

Temperature (K)

b

+ 90o

FIGURE 8.11: Intensity maps of (Qh, ´1.5, ´2Qh) reciprocal plane at different tempera-tures, where ´0.30 ă Qh ă ´0.20. Magnetic reflections of AF1, AF2 and AF5 magneticphases are visible in this cut of the reciprocal space. Each mapping has a number whichrefers to the temperature at which it has been measured, marked in the susceptibilitygraph.

The strong peak-overlap is the reason why AF1 and AF4 are the only phasesfor which details were obtained by collecting magnetic reflections. The AF4 phasewas studied at two temperatures, 11.5 K and 3 K, while data corresponding tothe AF1 phase were only collected at 9 K. The refined parameters that describethose magnetic structures are summarized in Table 8.3 and they were obtainedfrom the least square fitting of about 60 reflections of the AF4 phase and about210 of the AF1 one. The spin arrangements are depicted in Fig. 8.12 togetherwith their correspondent agreement plot. The AF4 phase of the 15%-Co-dopedcompound is similar to the one of the CoWO4, in the sense that the momentsare antiferromagnetically coupled along the a axis and they are located withinthe ac plane. Nevertheless, due to the experimental configuration (the samplewas mounted in a cryomagnet), only the l = 0 plane was available, and there-fore the treatment of these data yields two undistinguishable solutions that areθu = 32(2)˝ and θu = ´32(2)˝, which means that the moments are 58(2)˝ from atowards c or -c. This occurs because, as explained in the introductory chapters,magnetic neutron-diffraction is sensitive to the perpendicular component of the

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118CHAPTER 8. MAGNETIC STRUCTURE DETERMINATION AND FERROELECTRIC PROPERTIES OF

MN1´XCOXWO4 (X = 0.05, 0.10, 0.15 AND 0.20)

Magnetic phase (AF4) (AF4) (AF1)Temperature T = 11.5 K T = 3 K T = 9 Kk ( 1

2 , 0, 0) ( 12 , 0, 0) ( 1

2 , 14 , 1

4 )

Mn1 �(m)(μB) 2.08 (2) 1.14 (1) 3.67 (2)φu 0˝ 0˝ 0˝θu 32 (2)˝ or -32 (2)˝ 32 (2)˝ or -32 (2)˝ 142 (2)˝

Mn2 u2 = u1 u2 = u1 u2 = u1

Δϕ 0 0 0

RF2/% 11.9 15.3 21.9RF2w/% 11.3 12.7 15.6RF/% 10.7 14.4 30.9χ2 8.78 2.93 3.60

TABLE 8.3: Parameters that describe the AF1 and the AF4 commensurate magnetic struc-tures of Mn0.85Co0.15WO4 at different temperatures.

magnetic moment with respect to the scattering vector. As the crystal was mountedwith c axis vertical, and only l = 0 plane was accessible, both θu and ´θu anglesexplain the data. However, as magnetic measurements show that the easy axis islocated about 55˝ from the a axis toward -c, one can conclude that the real valueof θu is -32(2)˝=148(2)˝. The solution is depicted in Fig. 8.12(a). The amplitude ofthe moments decreases with decreasing temperature as revealed by the refinementat 3 K.

Regarding the AF1 phase, it is pretty similar to the AF1 phase of the pureMnWO4 compound [87]. The structure is collinear in the ac plane and the magneticmoments are arranged in a + + ´´ configuration. Unlike the pure compoundwhere moments are about 35˝ far from a towards c, in Mn0.85Co0.15WO4 spins arelocated at θu = 142(2)˝ from the c axis (52˝ from a, towards -c), see Fig. 8.12(b).

These results obtained for Mn0.85Co0.15WO4 give additional details comparedto the model proposed by Song et. al. [125] from powder samples with the samenominal composition and agrees quite well with the results obtained by Chaud-hury and co-workers in Ref. [127]. However, there are some discrepancies: con-cerning the transition temperatures to the AF2 and AF1 phases, Chaudhury et. al.ascribe a single temperature whereas in our data there is a narrow, but clear, tem-perature gap between both transitions (T2 and T3). Besides that, decreasing tem-perature, the intensity of the AF1 peak does not vanish in their sample whilst inours it does. In addition, the intensity of the AF2 peak keeps on increasing whereas

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8.3. MN0.85CO0.15WO4 119

0 1 2 30

1

2

3

(Fca

lc)2 (a

rb. u

nits

)

(Fobs)2 (arb. units)

T = 9 KAF1

RF = 30.9%RF2w = 15.6%

0 2 4 6 80

2

4

6

8

RF = 10.7%RF2w = 11.3%

T = 11 KAF4

(Fca

lc)2 (a

rb. u

nits

)

(Fobs)2 (arb. units)

0 2 4 6 80

2

4

6

8

RF = 14.4%RF2w = 12.7%

T = 3 KAF4

(Fca

lc)2 (a

rb. u

nits

)

(Fobs)2 (arb. units)

(a)

(b)

FIGURE 8.12: Magnetic models associated with the commensurate wave vectors inMn0.85Co0.15WO4: (a) Magnetic model for the AF4 magnetic phase with k = ( 1

2 , 0, 0).(b) AF1 spin arrangement with k = ( 1

4 , 12 , 1

2 ). The observed intensities compared to calcu-lated intensities with parameters given in Table 8.3 are shown next to each model.

our data does not show such evolution. All these differences between our resultsand those reported in Ref. [127] can be related to the strong frustration in the com-pound, and show how small modifications (such us for example tiny variations ofthe cobalt concentration around its nominal value) can derive in somewhat distinctresponses.

In summary, neutron diffraction has suggested the high degree of magneticfrustration of Mn0.85Co0.15WO4 as evidenced by the existence of several coexistingmagnetic phases. As the temperature decreases the collinear structure associatedwith k4 arises (characteristic for CoWO4 parent compound), but with further de-creasing the temperature new magnetic reflection related to k1, k2 and k5 emergeand compete.

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120CHAPTER 8. MAGNETIC STRUCTURE DETERMINATION AND FERROELECTRIC PROPERTIES OF

MN1´XCOXWO4 (X = 0.05, 0.10, 0.15 AND 0.20)

8.3.3 Electric polarization

Figure 8.13 presents the polarization along the a and b axes measured on cooling.Polarization along b arises at about 10.0(1) K, has its maximum at 9.3(1) K and dis-appears at about 8.5(1) K. This temperature range coincides with the one where theAF2 phase exists (on cooling) and confirms that its propagation vector describesthe periodicity of the AF2 multiferroic phase instead of the paraelectric AF3’s. Incontrast with the lower doped compositions, Mn0.85Co0.15WO4 exhibits anothermultiferroic phase between 6.0(1) K and 3.0(1) K with maximum polarization alongb at 3.9(1) K. The polarization in this range must come from the reappearance of theAF2 magnetic phase. The continuous increase of the polarization below 3.0(1) K isstill unclear. Note that the polarization along a undergoes the same accidents asPb, but in a smaller scale, that can be due to a small misalignment of the sample.

2 4 6 8 10 120

4

8

12

~T5~T4

~T3

P (μ

C/m

2 )

Temperature (K)

~T2

Pb

Pa

FIGURE 8.13: Temperature dependence of spontaneous electric polarization along theprinciple crystallographic directions measured while the temperature was decreasing.

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8.3. MN0.85CO0.15WO4 121

90 135 180 225 270

1.3

1.4

1.5

1.6

H = 1 kOe

25 K 5 K

+90o

-a-c

a

χ (1

0-4 C

GS)

Angle (o)

-100 0 100 200 300 4000

1

2

3H = 1 kOe

b+90o

Temperature (K)

χ-1 (1

05 CG

S)

0 5 10 15 20 25 301.2

1.3

1.4

1.5

1.6

1.7

H = 1 kOe

b

Temperature (K)

+90oχ (1

0-4 C

GS

)

TN2 TN

(a)

(b)

(c)

FIGURE 8.14: (a) Temperature dependence of susceptibility along the b axis and alongtwo particular directions within the ac plane, namely for field applied at an angle α ofabout 37˝ with respect to the c axis and for field perpendicular to it; (b) angular depen-dencies of susceptibility in H = 1 kOe within the ac plane, at 25 K and 5 K; (c) temperaturedependence of the inverse susceptibility.

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122CHAPTER 8. MAGNETIC STRUCTURE DETERMINATION AND FERROELECTRIC PROPERTIES OF

MN1´XCOXWO4 (X = 0.05, 0.10, 0.15 AND 0.20)

8.4 Mn0.80Co0.20WO4

8.4.1 Bulk magnetic response

Figure 8.14(a) presents the temperature variation of dc susceptibilities along theb axis and along two particular directions within the ac plane, namely for fieldapplied at an angle α of about 143˝ with respect to the c axis (named α axis) andfor field perpendicular to it. For those two orientations it was found that the sus-ceptibilities at „ 20 K have the highest and the lowest values, respectively. Thisis confirmed by the angular dependencies of the susceptibility measured in mag-netic field of 1 kOe rotating within the ac plane, shown in Fig. 8.14(b). At 25 K,above TN , the angular dependency of the magnetic susceptibility along the easy(α) and hard (α+90˝) directions exhibits its maximum and minimum, respectively,whereas below TN the situation is inverted. The difference of the susceptibilitiesin Figs. 8.14(a) and 8.14(b) is due to the small contribution from the rotator. Thesusceptibility along any other direction in the crystal was found to be a superposi-tion of the three orientations presented in Fig. 8.14(a). The peculiarity at 20.2(2) K,clearly visible in the susceptibility along all the directions, could be naturally at-tributed to the Néel temperature, TN . Another susceptibility anomaly at 8.6(3) K,which is evident for the two orientations perpendicular to the α axis, evidences achange in the magnetic structure. Within the experimental resolution no unam-biguous fingerprint of this transition was detected in the susceptibility along theα direction.

As seen from Fig. 8.14(c), the anisotropy observed at TN persists in theparamagnetic region with all the three inverse susceptibilities becoming parallelstraight lines at high enough temperatures where the Curie-Weiss law applies.This results in significant differences in the asymptotic paramagnetic tempera-tures, ΘP, obtained for the three directions: ΘP = -70(2) K along the b axis,ΘP = -48(2) K for the α orientation, and ΘP = -83(2) K for the orientation perpen-dicular to α within the ac plane. Within the experimental errors, the effective mag-netic moment was found to be the same for all the orientations 5.78(5) μB, a valuewhich is somewhat higher than the one expected from the Mn2+ ions and the spinonly contribution from the Co2+ ions (5.57 μB) and thus, suggests that the Co or-bital moment is not completely quenched [43–45].

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8.4. MN0.80CO0.20WO4 123

8.4.2 Neutron diffraction

Below the Néel temperature, TN = 20 K, two magnetically ordered phases werefound, one of which is characterized by a single propagation-vector, k1, and theother one by two vectors, k1 and k2. The centering of 20 reflections at 2 K con-cluded that k2 = [-0.211(1), 1

2 , 0.452(2)] is an incommensurate propagation vector,while on the contrary k1 = ( 1

2 , 0, 0) is commensurate from TN down to the low-est temperature. Figure 8.15(a) shows the evolution of two magnetic reflectionscharacteristic of k1 and k2 with the temperature. When decreasing temperature,below 20 K the reflection related to k1 appears. Its intensity increases down to8.5 K where there is a change in its evolution. At this point new reflections asso-ciated to k2 emerge. With further decreasing of the temperature, the intensity ofthe latter reflection increases whereas the intensity of the commensurate reflectionslightly decreases.

Data collections were made in order to determine the magnetic structuresat 2 K and 11 K. The parameters of the obtained magnetic structures are given inTable 8.4, the spin arrangements are plotted in Figs. 8.15(b) and 8.15(c) and theagreement plots are depicted below each magnetic model. The magnetic structureat 11 K is a collinear one, the so-called AF4 [125, 127]. The moments lay in theac plane, along the easy-axis α that makes an angle θu = 142(2)˝ with the c˚ axis,see Fig. 8.15(b). In the collinear AF4 order, the magnetic moments in the zigzagchains of octahedra running along the c axis are ferromagnetically coupled. Chainsform ferromagnetic layers which extend parallel to the bc plane, and the adjacentlayers along the a axis are antiferromagnetically coupled. The ordered momentobserved at 11 K is 3.17(7) μB per magnetic atom.

The same set of reflections related to k1 revealed no difference (within the er-ror bars) in the magnetic component associated to k1 at 2 K. However at that tem-perature there is an additional propagation vector and new magnetic reflections ofthe type (hkl) ˘ k2 arise. The refinement of the 212 latter type of reflections let usconclude that the incommensurate part is an elliptical cycloid, the eccentricity ofwhich is ε = 0.61 according to the definition given in equation 8.1. The main axes ofthe cycloid are along the b axis and a direction in the ac plane that makes and angleω = 55(1)˝ with the c˚ axis (ω = α + 90˝). Therefore, the incommensurate part of themagnetic structure is rotating in a plane (bω) which is perpendicular to α (directionof the moments of the collinear component). The coexistence of two propagationvectors can thus be explained by a conical structure, as sketched in Fig. 8.15(c). Be-low TN2, mconical = mAF2 + mAF4 at each Mn/Co site. At 2 K the two components

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124CHAPTER 8. MAGNETIC STRUCTURE DETERMINATION AND FERROELECTRIC PROPERTIES OF

MN1´XCOXWO4 (X = 0.05, 0.10, 0.15 AND 0.20)

0 5 10 15 20 25

0

50

100

150

200

250

TN2

(0 0 0)- k2

(0 -1 -1)+ k1

Inte

nsity

(arb

. uni

ts)

Temperature (K)

TN

0 2 40

2

4

RF= 7.41%RF2w = 9.11%

T = 11 K

(hkl) k1

(Fca

lc)2 (a

rb. u

nits

)

(Fobs)2 (arb. units)

0 2 40

2

4

(Fob

s)2 (arb

. uni

ts)

RF = 7.45 %

RF2w = 6.17 %

RF = 6.54 %

RF2w = 8.74 %

T = 2 K (hkl)

k1

(hkl) k2

(Fobs)2 (arb. units)

(a)

(b) (c)

c

a

c

a

FIGURE 8.15: (a) Integrated intensities of two magnetic reflections as a function of thetemperature (the error bars are smaller than the symbols). (b) and (c) show the magneticstructures of Mn0.80Co0.20WO4 at 11 K and 2 K, respectively. The corresponding agree-ment plots of the refinements are plotted below each model.

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8.4. MN0.80CO0.20WO4 125

are rather similar: mAF4 = 3.23 μB and 2.42μB ď mAF2 ď 3.05μB (since mAF2 is anelliptical helix). As a result, the maximum amplitude on the Mn/Co site is 4.44 μB.This value is practically the theoretical saturation moment expected [4.6 μB per(Mn,Co) site in Mn0.80Co0.20WO4, having both Mn2+ and Co2+ in high-spin state].

Temperature T = 11 K T = 2 K T = 2 KMagn. phase AF4 Conical Conicalk ( 1

2 , 0, 0) ( 12 , 0, 0) [-0.211(1), 1

2 , 0.452(2)]

Mn1 �(m)(μB) 3.15(2) 3.23(6) 3.05(3)φu 0˝ 0˝ 0˝θu 142(2)˝ 143(2)˝ 55.0(9)˝�(m)(μB) -2.42(4)φv 90˝θv 90˝

Mn2 ux,z2 = ux,z

1 ux,z2 = ux,z

1 u2 = ´u1

uy2 = uy

1 = v = 0 uy2 = uy

1 = v = 0 v2 = ´v1

Δϕ 0 0 kz2

RF2/% 8.90 8.37 6.86RF2w/% 9.11 8.74 6.17RF/% 7.41 6.54 7.45χ2 26.2 25.4 4.37

TABLE 8.4: Parameters that describe the magnetic structures of Mn0.80Co0.20WO4. Mag-netic atoms Mn1 and Mn2 are located in the crystallographic unit cell at [ 1

2 , 0.6847(5), 14 ]

and [ 12 , 0.3153(5), 3

4 ], respectively.

The alternative scenario that could also result in two coexisting propaga-tion vectors, namely the one based on phase separation with distinct volume frac-tions, can be ruled out since, according to the experimental intensities, the atomicmagnetic moment of one of the hypothetic phases would be unrealistically high.Indeed, if we assume that there are two equal volumes for the two phases, thecollinear and the cycloidal structures, VTot = 2Vk1 = 2Vk2 , the magnetic momentsshould be multiplied by

?2, resulting in a too large total moment at 2 K.

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126CHAPTER 8. MAGNETIC STRUCTURE DETERMINATION AND FERROELECTRIC PROPERTIES OF

MN1´XCOXWO4 (X = 0.05, 0.10, 0.15 AND 0.20)

2 4 6 8 100

10

20

30

40

AF4 (P = 0)

AF2 (Pb) + AF4

TFE

Pb (

μC

/m2 )

Temperature (K)

FIGURE 8.16: Temperature dependence of spontaneous electric polarization along b axisin E = 0. Data was collected while heating, after cooling the sample from the paraelectricstate down to 1.9 K in poling field Ep = 1.8 V/cm.

8.4.3 Electric polarization

Spontaneous electric polarization was observed only along the b axis. It arises be-low TFE « 8.2 K and increases up to 45 μC/m2 at 2 K as shown in Fig. 8.16. Theonset in the temperature dependence of the electric polarization corresponds tothe observed anomalies in the magnetic susceptibility along the b axis and per-pendicular to the easy axis at « 8.6 K [Fig. 8.14(a)] as well as the onset of theincommensurate (0 0 0)´k2 magnetic reflection [Fig. 8.15(a)]. Thus the observedconical spin structure can be also identified as a ferroelectric state with P‖b whichis likely driven by the inverse Dzyaloshinskii-Moriya magnetoelectric interaction.

At first glance the existence of the P‖b polarization makes the cycloid partof the composed conical phase similar to the cycloidal ferroelectric phase AF2 inpure and slightly doped (Co 5%) MnWO4 [87, 125, 126]. However, there is a prin-cipal difference between them. In Mn0.80Co0.20WO4 the cycloidal component ofthe composed spin structure is developed below TFE on top of the already existingand more stable collinear AF4 phase in which spins are oriented along the easydirection in the ac plane (142˝ with respect to the c˚ axis). So, the appearance ofthe competing incommensurate cycloid AF2 (k2) spin order along easy directionis suppressed at low temperatures and it occurs in the directions being orthogonalto the spins in the existing AF4 spin order. Thus the AF2 spin order can appearonly in the magnetically hard directions. This is clearly seen from our neutron

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8.5. X ´ T PHASE DIAGRAM 127

data revealing the main cycloid axes along b and perpendicular to the easy di-rection in the ac plane (ω axis at 55˝ from c˚ axis). This circumstance, i. e. thecompetition between the magnetic anisotropy and the isotropic exchange interac-tions stabilizing the cycloidal structure, could explain the observed decreasing ofthe ferroelectric transition temperature from « 12.5 K in pure MnWO4 down to« 8.5 K in Mn0.80Co0.20WO4 and likely its further suppression and disappearancewith increasing Co concentration.

8.5 x ´ T phase diagram

Summarized below are the features that we have found in the x = 0.05, 0.10, 0.15and 0.20 compositions of the Mn1´xCoxWO4 family:

˛ Mn0.95Co0.05WO4: Two magnetic transitions were detected by bulk magneticmeasurements at TN = 13.2(2) K and TN2 = 12.2(2) K that separate three dis-tinct magnetic phases: the paramagnetic, the AF3, and the cycloidal AF2phase. The two transition temperatures are very similar to TN2 and TN inthe pure composition (12.3 and 13.5 K, respectively). Both magnetic ordershave the same periodicity, k = [-0.216(1), 1

2 , 0.461(1)]. At this low cobalt dop-ing the collinear commensurate AF1 phase, characteristic of the ground statein pure MnWO4 [87], is suppressed. The stability range of the AF3 order isso narrow and the ordered magnetic moments so small that after several at-tempts it was difficult to ensure that the sample was in the AF3 phase duringthe neutron data-collection and also to safely ascribe a magnetic model to de-scribe the data. The AF2 magnetic structure is a cycloidal spin-arrangementin the ub plane, where u is within the ac plane about 13˝ from a (towardsc). We observed that whilst temperature decreases, the magnetic anisotropyin the ub plane effectively diminishes as indicated by the decrease of the ec-centricity of the elliptical envelope of the moments. This magnetic structurehas the same symmetry as the AF2 phase of MnWO4, and consequently hasa unique polar axis, the b axis. Accordingly, the emergence of the cycloidalstructure is linked to the appearance of the electric polarization along b.

˛ Mn0.90Co0.10WO4: Bulk magnetic measurements revealed two magnetic tran-sitions: TN = 13.2(1) K and TN2 = 11.4(1) K. As for Mn0.95Co0.05WO4 the AF3phase is present in the crystal. In the collinear model for the AF3 phase thespins are located within the ac plane, about 34˝ from a towards -c, in contrast

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128CHAPTER 8. MAGNETIC STRUCTURE DETERMINATION AND FERROELECTRIC PROPERTIES OF

MN1´XCOXWO4 (X = 0.05, 0.10, 0.15 AND 0.20)

to lower doped compositions. Our data refinement concluded that an addi-tional spin component along the b axis is also plausible in the AF3 phase ofthis composition, even if its amplitude is rather small. Below TN2, the multi-ferroic phase emerges, the symmetry of which differs from AF2-structure’s,and does not allow any polarization along b, but within the ac plane. There-fore, the multiferroic phase of Mn0.90Co0.10WO4 is named AF2’ instead ofAF2. The latter is confirmed by the polarization measurements that showboth Pa and Pc components of spontaneous electric-polarization below TN2

down to the lowest temperatures („ 2 K).

˛ Mn0.85Co0.15WO4: The magnetic behavior of this composition is the mostcomplicated among the studied ones. The numerous and complex magnetictransitions below TN = 16.7(2) K demonstrated the strong competition be-tween distinct magnetic phases. In this case, the easy axis is located withinthe ac plane about 58˝ from a towards -c. Due to the strong peak overlap,only two collinear structures were studied in detail: AF1 and AF4. The for-mer is similar to the AF1 of the pure compound but the moments are closerto c, close to the easy axis. Regarding the AF4 phase, it is a simple antiferro-magnetic structure along a with the moments in the ac plane parallel to themagnetic easy-axis. In the paramagnetic region there is magnetic anisotropyfound that is attributed to the substitution of Mn by the anisotropic Co.

˛ Mn0.80Co0.20WO4: This crystal displays a coexistence of commensurate-collinear AF4 and incommensurate-cycloidal AF2 structures, which producesa ferroelectric ground state characterized by a conical antiferromagnetic spinorder. The translational symmetry of the double-k magnetic structure is de-termined by two wave vectors k1 = ( 1

2 , 0, 0) and k2 = [-0.211(1), 12 , 0.452(2)].

First, the collinear AF4 phase (k1) (similar to pure CoWO4’s structure) ap-pears at TN « 20 K and then, decreasing temperature below TN2 « 8.5 K,this collinear phase coexists with the cycloidal AF2 spin structure (k2). Thissecond modulation is accompanied by the appearance of ferroelectric polar-ization along b axis. We have found that spins in the AF4 phase are orderedalong the easy direction in the ac plane, at an angle of 52˝ with respect tothe a axis (towards -c) and does not change down to the lowest temperaturereached (2 K). The collinear order is clearly reminiscent of the antiferromag-netic order in CoWO4 (AF4, TN = 55 K, α « 137˝) [116]. In the cycloidalAF2 structure observed below TN2 the spin ordering occurs perpendicular tothe easy-axis α of AF4, i. e. in magnetically hard directions that makes thisstructure different compared to the pure cycloidal AF2 phase at lower cobaltconcentrations. As in Mn0.85Co0.15WO4, the magnetic anisotropy found in

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8.5. X ´ T PHASE DIAGRAM 129

the paramagnetic region is attributed to the substitution of manganese bythe anisotropic cobalt.

The results obtained for the 5%-Co-doped crystal agree rather well with thedescription given by Song and co-workers in Ref. [126]. However, our resultsreveal more and new details. Following our superspace-formalism-backed idea foran alternative description of the AF3 structure of MnWO4, we propose for first timea cycloidal but antiferroelectric magnetic structure for the AF3 phase not only forthis 5%-Co composition but also for Mn0.90Co0.10WO4. In Ref. [126], the magneticstructure and the electric polarization of Mn0.90Co0.10WO4 were shown but, again,the details of the structure (u1,2 and v1,2 components of the magnetic moments)were not provided. Anyhow, their results are qualitatively consistent with ours.Regarding the Mn0.85Co0.15WO4, the phase diagram proposed by Song et. al. inRef. [125] ascribed just two magnetic phases to this material: one associated tok4 wave vector and the other one to the co-existence of k2 and the former. Onthe contrary the number of propagation vectors that we found is four (k1, k2, k4

and k5) coinciding with those detected by Chaudhury et. al. in Ref. [127], butnot with the five found by Feng and co-workers in Ref. [130]. Nevertheless, wehave seen up to five characteristic temperatures that could be ascribed to magnetictransitions instead of just three as observed in Ref. [127].

The recent work done by Feng et. al. [130] and Liang et. al. [131] considerthat the AF5 structure is the so-called AF2’ of the Mn0.90Co0.10WO4. The AF2’gives rise to the polarization within the ac plane. Though, in Mn0.85Co0.15WO4,where both k2 and k5 exist, there is not evident polarization within the ac plane.Nevertheless, they suggest neutron-diffraction experiments to really verify theirproposal. In the lack of any neutron data, we consider that the AF2’ and the AF5magnetic phases are independent. From our measurements the temperature rangefor the AF1 phase in the 15%-Co-doped sample is slightly bigger than that givenin Ref. [131]. The AF4 and AF1 phases of Mn0.85Co0.15WO4 have also been deter-mined in this work. Last but not least, the magnetic structures of Mn0.80Co0.20WO4

have been studied for the first time and the results have been corroborated by Fenget. al. in Ref. [130].

This work is limited to the study of just five family members of theMn1´xCoxWO4 solid solution. For that reason, the borders between magneticphases (in the x direction of the phase diagram) are not definitely established. Nev-ertheless, the phase diagram plotted in Fig. 8.18 shows the temperatures at whicheach composition undergoes a magnetic transition (red circles) obtained from our

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130CHAPTER 8. MAGNETIC STRUCTURE DETERMINATION AND FERROELECTRIC PROPERTIES OF

MN1´XCOXWO4 (X = 0.05, 0.10, 0.15 AND 0.20)

bulk magnetic data and the points at which the magnetic structures have beenrefined (black crosses). This information is on top of the x ´ T phase diagram pro-posed for Mn1´xCoxWO4 by Liang et. al. in Ref. [131].

8.6 Evolution of the easy axis

The evolution of the orientation of the easy magnetization directions within theac plane with Mn2+ substitution by the anisotropic Co2+ ions is summarized inFig. 8.17. One can observe how the increase of cobalt in the composition changesthe direction of the easy axis. In the extreme cases, i. e. in MnWO4 and CoWO4,the easy axis is located about 35˝ from a axis (towards c) and about 45˝ from a axis(towards -c) respectively, see Refs. [87, 116]. From our investigation we have seenthat just 15% of cobalt is enough to fix the easy axis about 10˝ from its location inthe pure CoWO4.

The observed finite values of the susceptibility along the easy directions of allthe compositions at low temperature differ qualitatively from those of the simplecollinear antiferromagnet (where parallel susceptibility vanishes at low tempera-ture) and are somewhat reminiscence of the susceptibility behavior in screw-typemagnetic structures [48–50].

x = 0 x = 0.20 x = 0.05 x = 1 x = 0.10 x = 0.15

a

-c

c

FIGURE 8.17: Magnetic easy-axes of the Mn1´xCoxWO4 family plotted in the ac plane.

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8.6.EV

OLU

TION

OF

THE

EASY

AX

IS131

Mn2 Mn1 b

a

Mn22 Mn11

0.0 0.1 0.2 0.30

4

8

12

16

20

24

AF3

F1

AF1

AF2’ or AF5AF2 AF4+AF2

Tem

pera

ture

(K)

x in Mn1-xCoxWO4

AF4

Paramagnetic

AAAAFFFF2222

Mn2 Mn1

Mn22 Mn11

c

a

c

a

c

a

AF1

b

a

FIGURE 8.18: Phase diagram proposed by Liang et. al. in Ref. [131]. Red circles (transition temperatures) and black crosses (magnetic structuredetermined) are our contribution. Circles crossed mean that there is coexistence of phases as found in our data.

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132CHAPTER 8. MAGNETIC STRUCTURE DETERMINATION AND FERROELECTRIC PROPERTIES OF

MN1´XCOXWO4 (X = 0.05, 0.10, 0.15 AND 0.20)

8.7 Evolution of the orientation of the electric polar-ization

Within the microscopic scheme of spin current or inverse Dzyaloshinskii-Moriyamodel, the expected electric polarization can be expressed as

P „ rij ˆ (Si ˆ Sj) (8.2)

where rij is the vector between two neighboring spins, Si and Sj. Note that, inthe multiferroic phase of the studied compounds, two neighboring atoms alongthe b axis do not contribute to the polarization because the spins are antiparalleland thus, Si ˆ Sj is zero. Lets then calculate the contribution of the coupling be-tween two neighboring Mn atoms in the chains (parallel to c), P1, and betweentwo neighboring Mn atoms in distinct chains parallel to a, P2, in the multiferroicAF2 phase. For this calculation it is assumed that the unit cell is orthogonal forsimplicity (β = 90˝). The orientation of each Mni magnetic moment in the unit cellcan be expressed in the following way (see the relation between mu and my in therefinement tables):

Case 1 Ñ Calculation within the chains along c

i = 1 Ñ S1 = m1u cos(2πk ¨ R)u + m1y sin(2πk ¨ R)y (8.3)

i = 2 Ñ S2 = m1u cos[2π(k ¨ R +kz

2) + π]u + m1y sin[2π(k ¨ R +

kz

2) + π]y

(8.4)

Case 2 Ñ Contribution of the interchain interaction along a

i = 1 Ñ S1 = m1u cos(2πk ¨ R)u + m1y sin(2πk ¨ R)y (8.5)

i = 2 Ñ S2 = m1u cos[2π(k ¨ R + kx)]u + m1y sin[2π(k ¨ R + kx)]y (8.6)

Taking into account that the direction u in the AF2 phase is within the ac plane(φ = 0) and that the vectorial product of the spins is parallel to the normal of therotation plane of the moments, n,

S1 ˆ S2 = |Si ˆ Sj|nCase 1 Ñ m1um1y sin(πkz)(sin θ i + cos θk) (8.7)

Case 2 Ñ m1um1y sin(2πkx)(sin θ i + cos θk) (8.8)

We will focus on Case 1. To calculate the electric polarization induced by the an-tisymmetric coupling of the spins, we have to take into account the r12 vector that

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8.7. EVOLUTION OF THE ORIENTATION OF THE ELECTRIC POLARIZATION 133

goes from Mn1 to Mn2

r12 = ´yj +c2

k (8.9)

P1 = C1m1um1y sin(πkz)(´yj +c2

k) ˆ (sin θ i + cos θk) (8.10)

= C1m1um1y sin(πkz)(´y cos θ i +c2

sin θ j + y sin θk) (8.11)

where Ci is related to the spin-orbit coupling. Note that the contribution of thenext Mn2 Mn11 couple in the chain will be

P1 = C1m1um2y sin(πkz)(´y cos θ i +c2

sin θ j ´ y sin θk) (8.12)

and therefore, the contribution of the Mn atoms within the zigzag chains is parallelto the b axis.

P1 ‖ b = C1m1um1yc sin(πkz) sin θ j (8.13)

On the other hand, the contribution of the interchain coupling results in

r12 = ai (8.14)

P2 = C2m1um1y sin(2πkx)(ai) ˆ (sin θ i + cos θk) (8.15)

P2 ‖ b = ´C2m1um1ya sin(2πkx) cos θ j (8.16)

In conclusion, the total electric polarization will come from the sum of bothP1 and P2, at least.

P = P1 + P2 = m1um1y(C1c sin(πkz) sin θ ´ C2a sin(2πkx) cos θ) j (8.17)

When the rotation plane gets close to the ac plane θ becomes smaller and hence,the electric polarization too, see Fig. 8.20(a). According to Sagayama et. al. inRef. [96] there is a critical value for θ that makes unstable the polarization along b,and therefore it flops to the ac plane, the so-called AF2’ phase (or AF5 in [130, 131]).This critical value, θc, corresponds to certain cobalt concentration xc, which shouldbe 0.05 ă xc ă 0.10 (xc = 0.075 according to Refs. [130, 131]).

If we consider that the n vector it is not restricted to the ac plane (sin θ cos φi+sin θ sin φ j + cos θk), then P1 and P2 have a more general expression:

P1 = C1m1um1yc sin(πkz)(´ sin θ sin φi + sin θ cos φ j) (8.18)

P2 = C2m1um1ya sin(2πkx)(´ cos θ j + sin θ sin φk). (8.19)

Note that for 10%-Co-doped composition n = b (θ = 90 and φ = 90) and thereforeP is constrained to the ac plane.

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134CHAPTER 8. MAGNETIC STRUCTURE DETERMINATION AND FERROELECTRIC PROPERTIES OF

MN1´XCOXWO4 (X = 0.05, 0.10, 0.15 AND 0.20)

2 4 6 8 10 12 140

25

50

75

100

Pa

Mn0.90Co0.10WO4

P (μ

C/m

2 )

Temperature (K)

(b)Pc

2 4 6 8 10 12 140

15

30

45

x = 0.15

x = 0.2

x = 0.05P

b (μC

/m2 )

Temperature (K)

(a)Mn1-xCoxWO4

x = 0

FIGURE 8.19: Electric polarization of the x = 0, 0.05, 0.10, 0.15 and 0.20 compositions ofthe Mn1´xCoxWO4 family.

In Refs. [130, 131] it is suggested that once the polarization is within theac plane (AF2’ or AF5), with increasing the concentration of cobalt the rotationplane of the moments rotate continuously around the a axis so that the electricpolarization does it too (θ ă 90˝ or/and φ ă 90˝). Hence, Pa and Pc decreasewith increasing x. However, if it was so, the polarization along b should increaseand the latter is not observed, which is attributed to an accidental cancelation ofthe contribution to Pb coming from P1 and P2. Anyhow, according to Feng andLiang the decrease of the Pa and Pc components, again makes the AF5 phase un-stable flipping the electric polarization towards b at concentrations of cobalt abovexc2 „ 0.135. These two groups locate the x = 0.15 compound just in the boundary,and therefore they consider that the AF5 phase that appears should give rise to

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8.7. EVOLUTION OF THE ORIENTATION OF THE ELECTRIC POLARIZATION 135

very small (even not measurable) electric polarization within the ac plane, assum-ing that θ ăă 90˝ or/and φ ăă 90˝.

0.0 0.1 0.2 0.30

4

8

12

16

20

24

PE

Paraelectric or antiferroelectric

Pa+ PcPb Pb

Tem

pera

ture

(K)

x in Mn1-xCoxWO4

Paraelectric (PE)

FIGURE 8.20: Phase diagram of Mn1´xCoxWO4 taking into account the electric order,based on the phase diagram proposed by Liang et. al. in Ref. [131] to locate the bound-aries.

From our point of view, it is not clear the AF2’ and AF5 phase are the same.The point symmetry of the AF2’ phase (present in Mn0.90Co0.10WO4) is m11, sincethe data can be refined using the superspace group Xc11(α0γ)ss that arises fromthe superposition of two mG2 magnetic modes, see Appendix E. This phase is notallowed to give rise to macroscopic electric polarization along the b axis. So, ifthe rotation plane of the spins would rotate around a as suggested by Liang inRef. [131], the polarization along b must be compensated for every θ and φ valuesin equations (8.18) and (8.19). Otherwise, the coexistence of polarization along a, band c axes requires either a phase separation, or the point symmetry 111, situationin which the number of parameters free for the refinement would be doubled.Then, if there is any rotation of the plane as suggested by Liang et. al. [131] thesymmetry of the system would change and consequently the multiferroic AF2’phase and AF5 phases would not be equal.

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136CHAPTER 8. MAGNETIC STRUCTURE DETERMINATION AND FERROELECTRIC PROPERTIES OF

MN1´XCOXWO4 (X = 0.05, 0.10, 0.15 AND 0.20)

8.8 Summary and conclusions

We have contributed to precise the x ´ T phase diagram of Mn1´xCoxWO4. De-tailed descriptions of the magnetic structures of several concentrations have beenprovided together with a careful study of the bulk magnetic properties and theelectric polarization. Our main contributions to the x ´ T phase diagram are:

(i) Complete description of all the investigated magnetic structures, giving theFourier components of moments of the magnetic atoms in the unit cell.

(ii) New cycloidal model which describes the AF3 phase.

(iii) Evolution of the cycloidal phase with decreasing temperature in x = 0.05 and0.10: the eccentricity and the amplitude of the mayor and minor semi-axes.

(iv) Detection of the phase transitions of x = 0.15 and the low-temperature hys-teresis.

(v) Antiferromagnetic-conical structure proposed for the first time in this familyfor the x = 0.20 composition.

(vi) Based on the symmetry considerations, we claim that the AF2’ phase and theAF5 phases are not equal.

Therefore, we have confirmed the existence of new multiferroic phases vary-ing the cobalt content, which give rise to electric polarization either along b axis orwithin the ac plane.

Apart from the contributions to the x ´ T diagram, it has been found that theorientation of the easy-magnetic axis is close to the pure CoWO4’s one with just15% of cobalt. At this doping the AF4 phase (proper of CoWO4) is already verystable, however at low temperatures competing phases emerge and coexist due tohigh degree of magnetic frustration. Indeed, we have observed that increasing thecobalt content increases the magnetic frustration, being 15%-cobalt-doped compo-sition the most frustrated one of the series. Though, with further increasing theamount of cobalt, x ą 0.15, the frustration decreases.

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Chapter 9Lattice anomalies at the

ferroelectric and magnetictransitions in cycloidal

Mn0.95Co0.05WO4 and conicalMn0.80Co0.20WO4 multiferroics

I n this chapter we present a study of the thermal evolution of the cell pa-rameters and a description of the lattice anomalies detected in the vicinity ofthe successive magnetic transitions of two multiferroic Mn1´xCoxWO4 com-

pounds (x = 0.05 and 0.20).

These two systems have been chosen for the study because they have differ-ent magnetic ground states, unlike magnetic transitions, but similar ferroelectricstate. We have observed remarkable differences in the thermal evolution of the lat-tice parameters below 30 K. The comparison with the magnetic features providesevidence of distinct directional effects depending on the characteristics and sym-metry of the magnetic order parameters at the paraelectric-ferroelectric transition.

Synchrotron X-ray powder-diffraction data were collected at ultra-high reso-lution ID31 diffractometer of the European Synchrotron Radiation Facility (ESRF,Grenoble). To reduce the absorption, a short wavelength was selected(λ = 0.40000(1) Å). The finely grounded samples were enclosed in a 0.5 mm di-

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138CHAPTER 9. LATTICE ANOMALIES AT THE FERROELECTRIC AND MAGNETIC TRANSITIONS IN

CYCLOIDAL MN0.95CO0.05WO4 AND CONICAL MN0.80CO0.20WO4 MULTIFERROICS

ameter borosilicate glass capillary, and a good powder averaging was ensured byappropriate spinning of the capillary in the beam. To collect low temperature pat-terns the capillary was placed in a continuous liquid-helium flow cryostat withrotating sample rod. Diffraction patterns down to 5 K were recorded at differenttemperatures (on cooling) after appropriate waiting time for a good thermal stabi-lization. The counting time was about 1.5 hours to have the desired statistics overthe angular range 4˝ to 42˝ degrees in 2θ.

9.1 Lattice evolution in Mn0.95Co0.05WO4

As it has been discussed in Chapter 8, magnetic susceptibility measurements per-formed in Mn0.95Co0.05WO4 single crystal [plotted in Fig. 9.1 for the field appliedalong b axis] indicate only two transitions (at TN = 13.2 K and TN2 = 12.2 K),which separate three distinct magnetic phases: paramagnetic (PM), AF3, and thecycloidal AF2. The two latter with k = [-0.216(1), 1

2 , 0.461(1)]. We have observedthat at this low Co doping the collinear commensurate order (AF1) characteristicof the ground state in pure MnWO4 is suppressed and that the transition tempera-tures in Mn0.95Co0.05WO4 are very similar to TN2 and TN in the pure composition(12.3 K and 13.5 K, respectively).

The low temperature cycloidal phase of Mn0.95Co0.05WO4 displays the samecharacteristics as AF2 phase in MnWO4. The main difference between AF2 inMn0.95Co0.05WO4 and MnWO4 is the decrease of the angle between the easy planeand a axis in the former, see either Chapter 8 or Ref. [126]. Namely, the cycloidalplane is parallel to the b axis and the angle between the plane and the a axis is ofabout 15˝ in Mn0.95Co0.05WO4, smaller than in MnWO4 (« 34˝). Two projectionof the cycloidal order are depicted in Fig. 9.1. Remember that the ground stateof the 5%-Co sample is ferroelectric with the electric polarization along b axis, aspredicted by the point symmetry of the superspace group, see Chapter 8.

Figure 9.2(a) depicts the evolution of a, b, and c parameters of the 5%-Cocrystal obtained in the profile matching. As seen in Fig. 9.2(a), as temperaturedecreases approaching the transition into the incommensurate AF3 order, a mag-netoelastic anomaly appears in the paramagnetic region at the vicinity of the tran-sition temperature of TN (« 13.3 K), as signaled by the small but clear downturn inthe evolution of a, b, and c. These parameters suddenly contract below 14 K, appar-ently in a rather isotropic manner. Note that this spin induced strain disappears ex-actly at TN , and cell parameters sharply relax at entering of the AF3 magnetic order,

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9.1. LATTICE EVOLUTION IN MN0.95CO0.05WO4 139

c

a

b

a

0 2 4 6 8 10 12 14 16 18

1.4

1.5

1.6 H = 1 kOe TN2

Temperature (K)

χ (1

0-4 C

GS

)

b

+ 90o

TN

12 13

1.55

1.60

TN2 T

N

FIGURE 9.1: Magnetic susceptibility of Mn0.95Co0.05WO4 single crystal (cooling and heat-ing cycle, 1 kOe), measured with the magnetic field applied along a, b and c axes. Themultiferroic structure present below TN2 is projected along b and c axes.

recovering approximately their (larger) values. From data in Fig. 9.2(a), the spin in-duced changes in cell parameters correspond to Δa « Δc « ´1.3(4) ˆ 10´4 Å andΔb « ´1.7(4) ˆ 10´4 Å. Despite the fact that, within our experimental accuracy,these lattice anomalies associated to the order-disorder magnetic transition lookrather similar along the three axes, a certain degree of anisotropy in the observedlattice strain cannot be discarded. Decreasing further the temperature no cleareffects on the lattice were detected across TN2 (« 12.5 K) or down to the lowesttemperature.

The evolution of β(T) (the angle characteristic of the monoclinic P2/c cell)is shown in Fig. 9.2(b) together with the behavior of the unit cell volume. Theangle β in this compound evolves linearly in the temperature range shown in thefigure, increasing with temperature at a rate 2.4 ˆ 10´4˝/K. Therefore, within ex-perimental errors, we do not appreciate any anomaly or deviation related to thesuccessive magnetic phases. An anomalous volume contraction within the para-magnetic regime precedes the AF3 spin order. Concomitant with the onset of thismagnetic order, the strain energy accumulated in the compressed cell is releasedand the volume abruptly expands to achieve again its normal size. Finally, wehave not detected apparent volume anomalies on going from the AF3 to the AF2phase, at TN2. In the AF2 phase (below TN2) Fig. 9.2(a) shows that thermal con-

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140CHAPTER 9. LATTICE ANOMALIES AT THE FERROELECTRIC AND MAGNETIC TRANSITIONS IN

CYCLOIDAL MN0.95CO0.05WO4 AND CONICAL MN0.80CO0.20WO4 MULTIFERROICS

4 6 8 10 12 14 1690.996

90.997

90.998

90.999

91.000

Temperature (K)

β (o )

138.23

138.24

TN2

(b)

Vol

ume

(A3 )

TN

4 6 8 10 12 14 164.8159

4.8160

4.8161

4.9917

4.99184.9919

Cel

l par

amet

ers

(A)

Temperature (K)

(a)

5.7510

5.7512

a

c

TN2 TN

b

FIGURE 9.2: Evolution of the cell parameters of Mn0.95Co0.05WO4 as a function of thetemperature: (a) a, b and c, (b) β and unit cell volume.

traction on cooling is bigger for a and c axis than for b axis. This difference isreminiscent of the distinct behavior of b reported from thermal expansion mea-surements on MnWO4 by Chaudhury et. al. [95]: they observed a weak negativeexpansion of b between TN1 and TN . But the evolution of β was not determinedand its contribution to these thermal expansion data obtained using a capacitancedilatometer is unknown. In MnWO4 these authors reported a variation of the ex-pansion coefficients at TN and anisotropic step-like changes at TN1 = 7.5 K [95](Δa/a = ´3 ˆ 10´5, Δc/c = 1 ˆ 10´5, Δb/b = ´1 ˆ 10´5 at the entering of thereentrant paraelectric AF3, which has been suppressed in Mn0.95Co0.05WO4).

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9.2. LATTICE EVOLUTION IN MN0.80CO0.20WO4 141

9.2 Lattice evolution in Mn0.80Co0.20WO4

Increasing the cobalt content from x = 0.05 to 0.20 in MnxCo1´xWO4 producesremarkable changes in the magnetic and lattice behavior of the system. Accord-ing to the discussion in Chapter 8, the dc susceptibility of the Mn0.80Co0.20WO4

single crystal along the b axis (χb), plotted in Fig. 9.3(a), signals two magnetictransitions at about TN = 20.3 K and TN2 = 8.5 K. A simple collinear antiferro-magnetic order with kAF4 = ( 1

2 , 0, 0) sets below the Néel temperature, the socalled AF4 structure, whereas at TN2 another incommensurate propagation vec-tor arises kAF2 « (´0.21, 1

2 , 0.45), similar to the one of the pure MnWO4 andMn0.95Co0.05WO4 composition. Accompanying the latter wave vector, sponta-neous electric polarization appears along the b axis, i. e. in the same directionas in the ferroelectric phase of Mn0.95Co0.05WO4 (below TN2 = 12.2 K). However, itis important to note that, increasing Co doping from x = 0 the polarization first re-orients from the b axis to the ac plane in vicinity of x = 0.10. With further doping itflips again to the b axis for x = 0.20. Namely at x „ 0.10 the cycloidal plane of spinsflops by 90˝ from parallel to perpendicular to b axis, and then for x „ 0.20 it returnsback to the b axis. In addition, an important difference between magnetic ordersin the ferroelectric phases of low (x = 0.05) and high (x = 0.20) doping states is thatin Mn0.80Co0.20WO4 the collinear commensurate kAF4 modulation (characteristicof the paraelectric phase at TN2 ă T ă TN) coexists with the cycloidal magneticmodulation kAF2 « (´0.21, 1

2 , 0.45) that appears below TN2. The coexistence ofthis two propagation vectors below TN2 produces a conical antiferromagnetic spinorder in Mn0.80Co0.20WO4 (Fig. 9.3).

The evolution of a, b and c parameters determined from profile matchingprocedure in this compound are shown in Fig. 9.4(a) and they exhibit a non-monotonous evolution below 30 K. Two separate small anomalies are perceivedcoinciding with the magnetic transitions at TN and TN2. These three parame-ters increase slightly on cooling across the paramagnetic-to-AF4 transition, andthe observed changes are very isotropic: Δa/a « 1.5 ˆ 10´5, Δc/c « 2.2 ˆ 10´5,Δb/b « 1.7 ˆ 10´5. Other changes were observed on further cooling across theferroelectric temperature TN2. There is a sudden change of slope below 9 K indi-cating a superior thermal contraction: a and b parameters decrease faster, whereasthe change in c seems smaller. Figure 9.4(b) plots both the volume of the unit cell inMn0.80Co0.20WO4 across the two magnetic transitions and the obtained evolutionof the monoclinic angle β(T). Small but clear magnetoelastic effects are apparentwhen looking at the evolution of the β angle and the cell volume. The characteristic

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142CHAPTER 9. LATTICE ANOMALIES AT THE FERROELECTRIC AND MAGNETIC TRANSITIONS IN

CYCLOIDAL MN0.95CO0.05WO4 AND CONICAL MN0.80CO0.20WO4 MULTIFERROICS

c

a

0 5 10 15 20 25 301.3

1.4

1.5H = 1 kOe

Temperature (K)

χ b (10-4

CG

S)

TN2 TN

c

a

FIGURE 9.3: Evolution of the cell parameters of Mn0.95Co0.05WO4 as a function of thetemperature: (a) a, b and c, (b) β and unit cell volume.

volume of the intermediate AF4 phase (in the interval « 10 ´ 20 K) is bigger thanin both the paramagnetic and the ferroelectric conical phases, respectively aboveTN and below TN2. So, a small (ΔV/V « 7.3 ˆ 10´5) but sharp volume expansiontakes place at TN (at the entering of the AF4 phase). By further cooling the extravolume of the intermediate phase seems to be released by appearing the conicalferroelectric phase.

9.2.1 Influence of the spiral plane on β at the ferroelectrictransition

The monoclinic distortion described by angle β decreases with increasing the cobaltcontent. The β value at 5 K is 90.9968(2)˝ for x = 0.05 and 90.8353(2)˝ for x = 0.20.The monoclinic angle shows a very different thermal evolution in these two multi-ferroics. They are compared in Fig. 9.5 below 30 K. Although the origin of the leftand right axes differs in the figure, the scale of both y axis is exactly the same. Theevolution above 10 K displays a very similar slope in both samples (2.3 ˆ 10´4˝/Kand 2.4 ˆ 10´4˝/K, respectively). Regarding Mn0.80Co0.20WO4 there are two visi-ble anomalies coinciding with the successive magnetic orders. The first is seen asa small step across TN (Δβ « ´1 ˆ 10´3˝ entering the AF4 phase). But more ap-

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9.2. LATTICE EVOLUTION IN MN0.80CO0.20WO4 143

4 8 12 16 20 24 2890.831

90.832

90.833

90.834

90.835

90.836

Temperature (K)

β (o )

137.04

137.05

137.06

137.07

(b)

Vol

ume

(A3 )

TN2TN

4 8 12 16 20 24 284.7910

4.7912

4.9844

4.9846

a

c

C

ell p

aram

eter

s (A

)

Temperature (K)

b

TNTN2

5.73925.73945.73965.7398

(a)

FIGURE 9.4: Evolution of the cell parameters of Mn0.80Co0.20WO4 as a function of thetemperature: (a) a, b and c, (b) β and unit cell volume.

pealing is the anomaly of β angle detected in Mn0.80Co0.20WO4 when it enters theincommensurate magnetic order below 9 K. In spite of the normal thermal contrac-tion (which acts decreasing β upon cooling), this angle suddenly starts increasingbelow TN2. So, Fig. 9.5 shows that β (5 K) ą β(25 K). In absolute terms the atyp-ical augment is small ( Δβ =0.0040(2)˝ in the interval 5 K - 12 K), but remarkablebecause this anomalous evolution should be forced by the exchange mechanismresponsible for the Pb polarization below 9 K, and was not observed in the ferro-electric AF2 phase of Mn0.95Co0.05WO4.

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144CHAPTER 9. LATTICE ANOMALIES AT THE FERROELECTRIC AND MAGNETIC TRANSITIONS IN

CYCLOIDAL MN0.95CO0.05WO4 AND CONICAL MN0.80CO0.20WO4 MULTIFERROICS

5 10 15 20 25 3090.831

90.832

90.833

90.834

90.835

90.836 Mn0.80Co0.20WO4

Mn0.95Co0.05WO4

Temperature (K)

β (o )

90.995

90.996

90.997

90.998

90.999

91.000

Vol

ume

(A3 )

FIGURE 9.5: Evolution of the β of both Mn0.95Co0.05WO4 and Mn0.80Co0.20WO4.

9.3 Summary and conclusions

We have observed distinct lattice anomalies associated to some of the magnetictransitions in two selected compositions. The anomalies detected were rather small(Δl/l « 10´5) but prove that the spin-lattice coupling in this family of materialsis weak. The observed anisotropy of the lattice strain is also weak. We did notobserve strongly anisotropic step-like changes of a, b and c parameters in any ofthe two compositions. In addition to changes of slope in the thermal evolution ofthe lattice, sudden modifications in the cell dimensions at some of the magnetictransitions were observed.

In Mn0.95Co0.05WO4, the three a, b and c parameters abruptly contract below14 K preparing the appearance of magnetic order. This anomalous volume con-traction within the paramagnetic regime precedes the paraelectric AF3 spin order.At TN = 13.2 K the spin induced strain energy accumulated in the compressed cellis released and the volume abruptly expands to adopt its normal size. In this com-pound we have not detected lattice anomalies on going from the AF3 to the AF2phase.

Distinct small but unambiguous anomalies were observed in the lattice evo-lution of Mn0.80Co0.20WO4 associated to the magnetic transitions at TN = 20.3 Kand TN2 = 9.1 K. In the first (at the paramagnetic-to-AF4 transition) there is anisotropic cell expansion on cooling. The volume of the intermediate AF4 phase (inthe interval « 10 ´ 20 K) is bigger than in both the paramagnetic and low tempera-

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9.3. SUMMARY AND CONCLUSIONS 145

ture phases. The temperature evolution of the monoclinic angle β in Mn0.80Co0.20WO4

completely differs from the monotonous behavior of β(T) in Mn0.95Co0.05WO4.Surprisingly, the monoclinic angle β in Mn0.80Co0.20WO4 systematically increaseson cooling below TN2.

The different behavior of the lattice found in the ferroelectric phases of thesetwo compounds seems related to their distinct magnetic structures. In the AF2structure of Mn0.95Co0.05WO4 the spiral plane contains the b axis, and it is closeto the a axis (about 15˝ to the a axis). The AF2 phase is developed from the AF3order [k « (´0.22, 1

2 , 0.47)], simply by a new magnetic component parallel to bwhich adopts the same modulation vector kAF2. The magnetic order in the ferro-electric phase of Mn0.80Co0.20WO4 is a conical antiferromagnetic order, composedof a cycloidal component [kAF2 « (´0.21, 1

2 , 0.45)] and a collinear AF4 componentperpendicular to it [kAF4 = ( 1

2 , 0, 0)]. The rotation plane of the cycloidal compo-nent is parallel to b and forms an angle of about 35˝ with the a axis, see Chapter 8.It is important to notice that in the ferroelectric phase of the x = 0.05 sample the cy-cloidal plane contains the easy magnetic axis (defined by the collinear modulatedAF3 order in this composition). However, for x = 0.20 the cycloidal plane is per-pendicular to the magnetic anisotropy axis (defined by the collinear commensurateAF4 order). The emergence of a cycloid perpendicular to the easy magnetic axisof Mn0.80Co0.20WO4, with a second propagation vector (incommensurate), and itsstrong interaction with the preexisting collinear commensurate AF4 componentparallel to the easy axis can explain the qualitatively different lattice anomaliesobserved at the ferroelectric transition in the two compositions. In particular, theobserved change at TN2 of the monoclinic angle β related to the non-diagonal (i.e. shift) uxz component of the deformation tensor could be associated with themagnetoelastic coupling which include contributions from isotropic exchange andanisotropic interactions. Among latter there are anisotropic terms proportional touxzLx

k4Lz

k4and uxzLx

k2Lz

k2, where Lx,z

k4and Lx,z

k2are x, z (a˚, c˚) components of the

order parameters in the AF4 and AF2 phases, respectively, which are determinedby linear combinations of the corresponding Fourier components of Mn/Co spinsin two sites of the crystallographic unit cell. Due to the orthogonality of Lk4 andLk2 in the combine (AF4+AF2) conical structure the products Lx

k4Lz

k4and Lx

k2Lz

k2

have different signs that could result in a bend of the β(T) at TN2. One cannotexclude the contribution to the β(T) anomaly of exchange magnetoelastic termssuch as uxzL2

k4and uxzL2

k2whose difference determines also the bend in the β(T)

at TN2. Further studies are required to clarify the role of different mechanisms ofthe magnetoelastic coupling.

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146CHAPTER 9. LATTICE ANOMALIES AT THE FERROELECTRIC AND MAGNETIC TRANSITIONS IN

CYCLOIDAL MN0.95CO0.05WO4 AND CONICAL MN0.80CO0.20WO4 MULTIFERROICS

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Part III

Magnetic-field-inducedtransitions

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Chapter 10Magnetic-field-induced transition

on MnWO4 with field appliedalong b axis

In this Part we will study the effects of the applied magnetic field on the mag-netic structures. We begin with the parent MnWO4. In particular, the mag-netic structure of the so called "X phase" with a flopped electric polarization

emerging from the zero-field cycloidal structure of MnWO4 at high magnetic fieldapplied along the b axis has been characterized.

The evolution of the magnetic and polar phase-diagrams of MnWO4 underapplication of magnetic field have been explored in number of studies includingneutron diffraction, ultrasound, polarization and magnetization experiments inpulsed high magnetic fields, see Refs. [103, 161–163]. This evolution naturallydepends on the direction in which the field is applied (note that the magnetic prin-ciple axes are different from the crystallographic ones). It includes new high fieldphases and reveals remarkable magnetoelectric effects.

The evolution of the polar AF2 phase under magnetic field applied alongthe b axis is perhaps of highest interest and has been the most studied part of thephase diagram [31, 99–102, 108, 161]. It has been found [99] that the magneticfield stabilizes the AF1 and AF3 phases at the expense of the polar AF2 phase andmost importantly that a new magnetic phase in which the spontaneous polariza-tion changes from b to a axis emerges in a narrow temperature range, at fields

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150CHAPTER 10. MAGNETIC-FIELD-INDUCED TRANSITION ON MNWO4 WITH FIELD APPLIED ALONG

B AXIS

higher than about 11 T, see Fig. 10.1. Though as soon as the demonstration of thismagnetoelectric effect took place, the determination of the magnetic structure ofthe new phase was anticipated [99], to the best of our knowledge up to now thishas not been done and the term "X phase" continues to be in use [161].

We have characterized the magnetic structure of the so called X phase andtried also to shed some light on how it emerges from the zero-field AF2 phase.Some intriguing features of the field-induced transition from the AF2 to the neigh-boring AF1 and AF3 phases will be also presented.

6 7 8 9 10 11 12 13 140

2

4

6

8

10

12

14 AF2 ’P // a

P = 0

P = 0pa

ram

agne

tic

AF3

AF1

colli

near

-ICM

μoH

(T)

Temperature (K)

elliptical spiralAF2

P // b

collinear-CM

H // b

FIGURE 10.1: Magnetic phase diagram in magnetic field applied along the b axis as deter-mined in this work (blue and dark circles are obtained from susceptibility data; rhombsfrom neutron diffraction) and from the polarization measurements reported in Ref. [99](grey quadratic symbols).

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151

0 2 4 6 8 10 12200

240

280

320

360

(0 1 1)- k2 obs.

(-111)- k2 obs.

(0 1 1)- k2 calc.

(-111)- k2 calc.

Inte

nsity

(arb

. uni

ts)

μ0H (T)

H ~ 11 T

T = 10.6 KAF2 AF2’

9 10 11 12 13 140

50

100

150

200

AF3AF2’

TN~13.7 K

T~11.3 K

(0 0 0)+k2

(0 1 1)- k2

Inte

nsity

(arb

. uni

ts)

Temperature (K)

T~10.4 K

H = 12 TH//b

AF1

0

50

100

150

200

250 H = 12 TH//b

AF1 T~10.4 K

(0 0 0)+k

1

Inte

nsity

(arb

. uni

ts)

(a)

(b)

(c)

FIGURE 10.2: Temperature dependence of the integrated intensities of magnetic satellitesat 12 T applied along the b axis. (a) commensurate reflection with k1 = ( 1

4 , 12 , 1

2 ) and(b) incommensurate reflections with k2 = [-0.221(1), 1

2 , 0.462(1)]; (c) Integrated intensitiesof magnetic satellites characteristic of the AF2 phase at 10.6 K as a function of magneticfield applied along b axis. Empty and full symbols are the experimental and the simulateddata, respectively.

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152CHAPTER 10. MAGNETIC-FIELD-INDUCED TRANSITION ON MNWO4 WITH FIELD APPLIED ALONG

B AXIS

The neutron-diffraction experiment on a single crystal of MnWO4 was per-formed on the CEA-CRG D23 diffractometer.

In order to explore the phase boundaries at high magnetic field three mag-netic reflections were followed as a function of the temperature at 12 T (max-imum field available): one commensurate reflection indexed as (0 0 0)+k1 withk1 = ( 1

4 , 12 , 1

2 ) and two incommensurate indexed as (0 0 0)+k2 and (0 1 1)´k2 withk2 = [-0.221(1), 1

2 , 0.462(1)]. Figures 10.2(a) and 10.2(b) show the evolution of thosemagnetic reflections as a function of temperature and reveal three magnetic tran-sitions. Upon increasing temperature, the first transition occurs at about 10.4 K,a point at which the intensity of the commensurate peak vanishes and incom-mensurate peaks emerge. At higher temperatures (about 11.3 K) the evolutionof the incommensurate peaks changes, signaling another transition. Finally, above13.7 K the intensity of all magnetic peaks becomes zero indicating the paramag-netic state. These magnetic transitions should be related to AF1, X and AF3 phases[99]. T = 10.6 K was chosen as an optimum temperature to collect data and de-termine the magnetic structure of the X phase. At this temperature, the integratedintensities of two magnetic reflections characteristic of AF2 and X phases were fol-lowed as a function of the magnetic field [Fig. 10.2(c)]. Both monitored peaksbehave in a different manner in fields above and below 11 T, pointing a transitionbetween the AF2 and X phases.

0.2 0.4 0.6

0

4

8

12

16 0 T 12 T

Inte

nsity

(arb

. uni

ts)

sinθ/λ

Nuclear reflectionsT = 10.6 K

FIGURE 10.3: Integrated intensities of nuclear reflections collected at 0 T and 12 T (10.6 K).There is not any evidence of structural change nor ferromagnetic contribution.

Nuclear reflections were collected at 10.6 K at both, zero- and 12 T fields.From the refinement of the nuclear data at zero field it was found that the crystal

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153

structure agrees with the one given in Ref. [87]. High field data does not unveil anychange within the experimental error (see Fig. 10.3), indicating that the magneticcomponent aligned parallel to the field is negligible or not reflected in the bunchof collected reflections.

Then, 39 magnetic reflections were centered at 12 T and 10.6 K. The refine-ment of the propagation vector confirms that the modulation of the high fieldphase corresponds to a slightly different k2 = [-0.221(1), 1

2 , 0.462(1)] compared tothe zero field k = [-0.216, 1

2 , 0.457]. The refined parameters are gathered in Table 10.1.The zero field structure (refined to check the data) agrees with the descriptiongiven by Lautenschlager et. al. in Ref. [87], see Fig. 10.4(a). The high field mag-netic structure is again an elliptical helix but oriented perpendicular to the externalfield, i. e. in the ac plane, as depicted in Fig. 10.4(c). The major semi-axis of thehelix remains along the magnetic easy-axis. Note that, for sake of simplicity, inthe above description of the magnetic structures we have omitted additional finecomponents determined by anisotropic (relativistic) interactions and allowed bythe symmetry according to analysis presented in the Chapter 7.

Temperature 10.6 K 10.6 K 10.6 K 11.8 KField (H ‖ b) 0 T 9.5 T 12 T 12 TMagnetic phase AF2 AF2 X AF3 (collinear)

k (-0.216, 12 , 0.457) (-0.216, 1

2 , 0.457) [-0.221(1), 12 , 0.462(1)] (-0.216, 1

2 , 0.457)

Mn1 �(m)(μB) -3.02(6) -3.22(3) -3.28(4) -2.77(2)φu 0 0 0 0θu 59(1)˝ 58(1)˝ 56(1)˝ 56(1)˝�(m)(μB) 2.56(6) 1.78(4) -1.23(4)φv 90˝ 90˝ 0θv 90˝ 90˝ 146(1)˝

Mn2 u2 = - u1 u2 = - u1 u2 = - u1 u2 = - u1

v2 = - v1 v2 = - v1 v2 = - v1 v2 = v1 = 0

Δϕ kz2

kz2

kz2

RF2/% 8.89 7.90 9.20 14.1RF2w/% 9.22 6.43 7.34 8.78RF/% 9.43 9.22 9.89 22.9χ2 11.4 4.64 4.67 2.17

TABLE 10.1: Parameters that describe the magnetic structures of MnWO4 at zero fieldand under field applied along b. Magnetic atoms Mn1 and Mn2 are located in the crystal-lographic unit cell in [ 1

2 , 0.68414(2), 14 ] and [ 1

2 , 0.3154(2), 34 ], respectively.

In order to shed light on the way the high field phase evolves from the AF2phase, data were collected at an intermediate field of 9.5 T. In this case, the struc-ture is qualitatively the same as in AF2, however the component along the b axis

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154CHAPTER 10. MAGNETIC-FIELD-INDUCED TRANSITION ON MNWO4 WITH FIELD APPLIED ALONG

B AXIS

0 2 40

2

4

RF= 9.43 %RF2w = 9.22 %

T = 10.6 KH = 0 TAF2

(Fca

lc)2 (a

rb. u

nits

)

(Fobs)2 (arb. units)

0 2 40

2

4T = 10.6 KH = 9.5 TAF2

RF= 9.22 %RF2w = 6.43 %

(Fca

lc)2 (a

rb. u

nits

)

(Fobs)2 (arb. units)

0 2 40

2

4

RF= 9.89 %RF2w = 7.34 %

T = 10.6 KH = 12 TX

(Fca

lc)2 (a

rb. u

nits

)

(Fobs)2 (arb. units)

(a) (b) (c)

b

a

c

a

FIGURE 10.4: Magnetic structures under (a) zero field; (b) 9.5 T; and (c) 12 T field appliedalong b axis at 10.6 K.

(mb) is considerably smaller than at zero field and the component along u is a littlebit bigger. Therefore, the eccentricity of the helix becomes bigger with increasingfield. The magnetic structure is sketched in Fig. 10.4(b). These results suggestthat the transition under field occurs in an abrupt manner, in the sense that themain effect of the field on the magnetic structure is that mb decreases with increas-ing field until a threshold value (achieved at about 10.5 T) where mb flops to theac plane. The evolution of the magnetic peaks with the field has been simulatedunder the former assumption and is reproduced in Fig. 10.2(c). Though we cannot completely rule out a rotation of the plane in a narrow field interval between9.5 T and the threshold field, if the transition would occur by a smooth rotation ofthe plane around the easy axis, the intensity of those peaks should first decreaseand then increase, which is at odd with our experimental data. The suggested

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155

way in which the high field phase evolves from the AF2 phase is reflected in thepolarization changes observed upon the transition. As seen from Fig. 10.5, thePb spontaneous polarization continuously decreases and Pa only slightly increasesup to at least 10 T, followed by a rapid change in both components in the vicinityof the threshold field. It is believed [108] that the transition is of first order. Theobserved continuous decrease of the Pb, which has to be proportional to a b-spincomponent, is in good agreement with the observed decrease of mb. The lack ofclear discontinuity in P (Fig. 10.5) might be due to a small misalignment betweenthe applied magnetic field and b axis. It is worth noting that, as demonstrated inRef. [101], the polarization after the transition into the high field phase should bezero unless the field is slightly tilted.

0 5 10 15 200

10

20

30

40

H = 10.5 T

Pa

μ0Hb (T)

P (μ

C/m

2 )

T = 10.5 K

Pb

0 5 10 15-5

0

5

10dP

/dH

(μC

/m2 T)

μ0H

b(T)

FIGURE 10.5: Polarization P along a and b crystallographic directions at 10.5 K as a func-tion of field applied along the b axis and corresponding derivatives (insert).

With respect to the effect of the field applied along b axis in the AF3 sinu-soidally modulated collinear structure, the refinement shows that this structurereminds basically invariant up to 12 T with respect to the zero field structure [87],with slightly more saturated magnetic moment.

Complimentary to the neutron diffraction study, ac-magnetic susceptibilityhas been recorded in superimposed dc-magnetic fields applied along the b axis.Though the maximum strength of the available dc-field (9 T) is below the thresh-old field (11 T), those bulk magnetic measurements helped us to further character-ize the evolution of the AF2 upon application of magnetic field. The incrementalsusceptibility (plotted for better appreciation either as a function of temperatureat a constant dc-field or as a function of dc-field at constant temperature) given

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156CHAPTER 10. MAGNETIC-FIELD-INDUCED TRANSITION ON MNWO4 WITH FIELD APPLIED ALONG

B AXIS

0 1 20

1

2

RF= 22.9 %RF2w = 8.78 %

T = 11.8 KH = 12 TAF3

(Fca

lc)2 (a

rb. u

nits

)

(Fobs)2 (arb. units)

c

a

FIGURE 10.6: AF3 magnetic structure at 11.8 K under 12 T field along b.

in Fig. 10.7 confirms that magnetic field stabilizes the AF1 and AF2 phases at theexpense of the polar AF1 phase, narrowing the temperature range in which thelater exists. While the susceptibility of the AF1 and AF3 phases changes smoothlyand almost linearly with the field (Fig. 10.7(b), the curves at 6.5 K and 13.5 Krespectively) the susceptibility of the AF2 phase evolves in a non linear manner.The high field susceptibility at temperatures from the upper part of the zero fieldregion of the AF2 phase (between about 11.5 K and 12.5K) cross the AF3 suscepti-bility, form a maximum and finally merge the AF3 susceptibility. This behavior iscompatible with AF2 to AF3 field-induced transition. On the other hand, the highfield susceptibility at temperatures from the lower part of the zero field region ofthe AF2 phase (between about 7 K and 9.5 K) merge the AF1 phase susceptibil-ity, reflecting field-induced transition from AF2 to AF1 phase. The correspondingtransitions determined as the point at which the incremental susceptibility mergethe corresponding AF1 or AF2 curves, Fig. 10.7, are plotted on the phase diagramshown in figure 10.1 for increasing/decreasing field. It should be noted that whileno measurable hysteresis was observed at the AF2 to AF3 transition, the field in-duced AF2 to AF1 transformation is clearly hysteretic implying that in this casea first order transition takes place. It is worth to point out the "open hysteresisloop" at 7 K (see the inset in Fig. 10.7). At this temperature the initial zero-fieldsusceptibility value is not recovered after magnetic field was increased beyond thefield needed for the AF2 to AF1 transformation and then reduced to zero. Furthercycling the field reflects in minor hysteresis loops reminiscent of magnetic mate-rial with martensite type structural phase transition. The observed peculiarity inthe susceptibility is consistent with the magnetoelectric memory effect attributedto the existence of small ferroelectric embryos [102, 164, 165] of the AF2 phase attemperatures below the transition into the non-polar AF1 phase. The origin of the

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157

above phase separation at this stage is uncertain and further studies are needed toclarify whether it is of structural or other types. The data extracted from Fig. 10.7together with the neutron diffraction data let us to construct the magnetic phasediagram presented in Fig. 10.1. Our results are in good agreement with previouslypublished polarization data [99] (also plotted) apart from the smaller field effecton the Néel temperature documented by us, and most importantly the low tem-perature tail of AF2 expanded into the AF1 phase. This tale could be regarded asa support for the existence of AF2 "embryos" leading to the previously reportedpolarization memory effect [102, 164, 165].

0 2 4 6 8 10

1.50

1.55

1.60

1.65

12.5 K

11.75 K

7.5 K

8.5 K

9.5 K

7 K

12 K

11.5 K

10.6 K

χb’

(x10

-4 e

mu/

g O

e)

6.5 K (AF1)

μ0H (T)

13 K (AF3)

0 2 4

1.50

1.55

7 K

6.5 K (AF1)

6 8 10 12 14

1.50

1.55

1.60

1.65

Temperature (K)

9 T

7 T

5 T

2.5 T0 T

χb’ (

x10-4

em

u/g

Oe)

(hac)b= 10 Oe

(a)

(b)

FIGURE 10.7: Incremental susceptibility plotted for better appreciation either (a) as afunction of temperature at constant dc-field or (b) as a function of dc-field at constanttemperature.

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158CHAPTER 10. MAGNETIC-FIELD-INDUCED TRANSITION ON MNWO4 WITH FIELD APPLIED ALONG

B AXIS

10.1 Summary and conclusions

To summarize, the X phase of the pure MnWO4 has been determined by neutrondiffraction experiments. The X phase has been named as AF2’ since it has the samesymmetry as the multiferroic phase of Mn0.90Co0.10WO4.

An abrupt transition has been proposed to describe the AF2-to-AF2’ transi-tion: a decrease of the eccentricity upon increasing field until the rotation plane ofthe spins flops perpendicular to the field.

The electric polarization aligned parallel to b in the AF2 phase, continuouslydecreases and Pa slightly increases, followed by a rapid change in both componentsin the vicinity of the threshold field. The observed continuous decrease of thePb, is in good agreement with the observed decrease of mb. Small misalignmentsbetween the applied magnetic field and the b axis can result in the lack of cleardiscontinuity in P.

Evidence for coexistence of the nonpolar collinear commensurate magneticphase, AF1, and the polar cycloidal structure, AF2, during the field-induced AF2-to-AF1 transition in MnWO4 is provided based on incremental-magnetic-susceptibility results. These data also suggest that a trace of AF2 domains ex-ist at temperature lower than the AF2-AF1 boundary of the zero magnetic-fieldphase-diagram (this observation could be regarded as a support for existence ofAF2 embryos being responsible for the recently published polarization memoryeffect).

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Chapter 11Magnetic-field-induced

transitions in Mn0.95Co0.05WO4

V arious magnetic-field-induced phase-transitions observed inMn0.95Co0.05WO4 will be presented in this chapter, depending on the di-rection of the magnetic field with respect to the crystal orientation, namely

along the b axis and along the particular direction α, the easy axis.

It is worth recalling the results obtained in Chapter 8 for this composition.It was concluded that Mn0.95Co0.05WO4 has two magnetic phases in zero field,the AF3 phase between 13.2 K and 12.2 K and the AF2 multiferroic phase below12.2 K. The AF2 phase is a cycloidal structure with the moments rotating withinthe bα plane, α being the easy-magnetic axis located at about 15˝ from a towards c.For the AF3 phase, it was concluded that most likely the structure corresponds toa cycloidal structure with opposite chirality of the magnetic moments in the unitcell.

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160 CHAPTER 11. MAGNETIC-FIELD-INDUCED TRANSITIONS IN MN0.95CO0.05WO4

0 2 4 6 8 10

1.20

1.35

1.50

1.65T = 1.8 Khac = 10 Oef = 333 Hz

+ 90o

χ’ (1

0-4 C

GS

)

μ0H (T)

b

(b)

b

a

b

𝜶 a

c

𝜶 + 90

(a)

FIGURE 11.1: (a) Sketch of the direction along which the field was applied with respectto the orientation of the magnetic moments. In purple the ellipse that envelops the mag-netic moments. (b) Real component of the ac susceptibility versus superimposed dc fieldalong the easy (α) and hard (α + 90˝) directions within the ac plane, and along the b axis,obtained at 1.8 K.

11.1 Probing field-induced transitions by incremental-magnetic susceptibility

To detect field-induced transitions in those magnetic phases, ac-susceptibility wasmeasured along b, α and α + 90˝1 directions at fixed temperatures and varying theamplitude of the dc-field which was superimposed in the same direction as theac-field. Figure 11.1(a) shows a sketch of the direction of the field with respect tothe spin arrangement projected along c and b axes. In the former the ellipse that

1As previuously defined in Chapter 8, α + 90˝ is the magnetic hard-axis which is located within theac plane 90˝ from α.

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11.1. PROBING FIELD-INDUCED TRANSITIONS BY INCREMENTAL-MAGNETIC SUSCEPTIBILITY 161

0 2 4 6 8

1.20

1.35

1.50

1.65

H // hac = 10 Oef = 333 Hz

(b)

χ’ (1

0-4 C

GS

)

μ0H (T)

Temperature (K) 1.8 11 12 12.25 12.5 14 4 6 8

1.35

1.40

1.45

0 2 4 6 8

1.20

1.35

1.50

1.65H // bhac = 10 Oef = 333 Hz

χ’ (1

0-4 C

GS

)

μ0H (T)

Temperature (K) 1.8 5 8 10 11 12.5 14

(a)

0 2 4 6 81.351.361.37

11 K

14 K

12.5 K

FIGURE 11.2: Real component of the ac susceptibility along the b and α axes versus su-perimposed dc field, at different temperatures in (a) and (b) respectively. The inset arezooms.

envelops the magnetic moments is plotted in purple and the b axis is drawn in red.In the projection along b, the ellipse is a line at about 14˝ away from a (the so-called α axis) and 90˝ far from it within the ac plane is located the α + 90˝ axis. Theevolution of the incremental susceptibility at 1.8 K shown in Fig. 11.1(b), revealsa field-induced transition along the b direction and suggests a beginning of suchtransition along α, while no indication of any transition is seen along the hard axisup to the highest applied field of 9 T.

The transition along b starts at fields as low as 3.5 T, indicated by the in-crease of the real component of the ac susceptibility in Fig. 11.1(b), and ends atabout 7 T. The susceptibility evolution suggests that this transition is completed in

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162 CHAPTER 11. MAGNETIC-FIELD-INDUCED TRANSITIONS IN MN0.95CO0.05WO4

two stages (and presumably composed of two consecutive transitions), evidencedby the susceptibility increase from the almost constant low field value, followedby an almost linear increase and then a decrease to an almost constant high fieldvalue. The higher the temperature is, the lower the fields needed to induce thetransition are and the narrower the transition itself is, see Fig. 11.2(a). The twoconsecutive stages seem to merge into a single one at temperatures above 8 K. Inthe AF3 phase, between about 12.1 K and 13.2 K, the peak in the ac susceptibilitycompletely disappears and the slope of the curves is similar to the slope at about2 T that coincides with the slope of the high-field phase of the AF2 structure. In theparamagnetic region (14 K), as expected, the ac-susceptibility is practically flat.

Field-induced transition takes place also along the α direction. As seen fromFig. 11.1(b), at 1.8 K, the susceptibility initially increases almost linearly up toabout 7 T followed by much faster increase at higher fields indicating the begin-ning of the transition, which however reminds uncompleted to the highest fieldavailable of 9 T. The susceptibility maxima observed on the 12 K, 12.25 K and12.5 K curves (AF3 phase region) should be related to the crossing from AF3 tothe paramagnetic region rather than to a spin-flop-type transition expected for theAF2 phase.

11.1.1 Incremental-magnetic susceptibility along intermedi-ate orientations between α and α + 90˝

The incremental susceptibility was also measured at intermediate orientations be-tween the b axis and the easy- and the hard-magnetic axes, see Figs. 11.3(a) and11.3(b) respectively. The crystal was cautiously oriented by hand using a protrac-tor and put back in the magnetometer for each measurement. Figure 11.3(a) showshow the transitions evolve when the field is applied further from b and closer to α.In such cases, higher fields are necessary to produce changes in the magnetic struc-ture compared to the case where H ‖ b, and the transitions are broader. Howeverthe two-stage shape is maintained until the field is about 60˝ from b, where the sec-ond stage fully vanishes. Then, the ac susceptibility strongly increases and formsa rather a sharp peak. As one approaches the α direction the maximum value ofthe susceptibility increases considerably as well as the critical field.

On the contrary, if the field is tilted towards the hard axis starting from b, thetransition is broadened very fast and also the two-stage shape disappears at anglesas low as 13˝ from the b axis [see 11.3(b)].

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11.1. PROBING FIELD-INDUCED TRANSITIONS BY INCREMENTAL-MAGNETIC SUSCEPTIBILITY 163

With the aim of clarifying the nature of these transitions neutron diffractionexperiments were performed applying the field along b and α. Furthermore, theeffect of the external field on the electric polarization was also investigated. Wewill first discuss the influence of applying the field along b, H ‖ b, and then wewill make a similar discussion when the field is applied along α, H ‖ α.

0 2 4 6 8 101.1

1.2

1.3

1.4

1.5

1.6

1.7 T = 5 Khac = 10 Oef = 333 Hz

13o

5o

+ 90o

χ’ (1

0-4 C

GS

)

μ0H (T)

b

(b)

0 2 4 6 8 101.1

1.2

1.3

1.4

1.5

1.6

1.7 T = 5 Khac = 10 Oef = 333 Hz

25o

60o22o

45o50o

65o

+ 90o

χ’ (1

0-4 C

GS

)

μ0H (T)

b

75o(a)

FIGURE 11.3: Real component of the ac susceptibility versus superimposed dc field at5 K, (a) at different angles between the easy direction within the ac plane and the b axis;and (b) at different angles between the hard direction within the ac plane and the b axis.In (a) and (b) the angles increase with the deviation from the b axis.

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164 CHAPTER 11. MAGNETIC-FIELD-INDUCED TRANSITIONS IN MN0.95CO0.05WO4

11.2 Continuous rotation of the magnetic structureand of the electric polarization with field alignedparallel to b

11.2.1 Magnetic order

The neutron experiment started by measuring two magnetic reflections on D23while an external magnetic field was applied along the b axis. The field increasedfrom 0 T to 10 T. The integrated intensities of those two reflections, depicted inFig. 11.4, exhibit changes in the field range 2.5 T - 7 T, which agrees with the field-induced transition observed by means of ac-susceptibility measurements with H}b.The (0 0 0)+k shows a evolution characterized by a maximum and a minimumvalue that occur at the fields where the (0 0 1)´k starts increasing and has a maxi-mum, respectively.

0 2 4 6 8 10

2.0

2.5

3.0 (000)+k

(001)-k

μ0H (T)

(Fob

s)2 (arb

. uni

ts)

T ~ 2 KH//b

80

90

100

(Fob

s)2 (arb

. uni

ts)

FIGURE 11.4: Integrated intensities of (0 0 0)+k and (0 0 1)´k reflections as a function ofthe external field applied along b at about 2 K.

Then, the centering of roughly 20 magnetic reflections at 5 T and 10 T showedthat the propagation vector does not vary with the application of such fields com-pared to the one found at zero field, k = [-0.216(2), 1

2 , 0.460(2)]. The magneticstructures were refined from the integrated intensities of about 130 reflections, at5 T (within the transition) and 10 T (above the transition). The parameters thatdescribe the proposed spin arrangements are gathered in Table 11.1, the corre-sponding structures are depicted in Figs. 11.5(a), 11.5(b), 11.5(c) and 11.5(d), to-gether with their corresponding observed intensities compared to the calculatedones. The data treatment concluded that the high-field structure (μ0H = 10 T) is a

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11.2. CONTINUOUS ROTATION OF THE MAGNETIC STRUCTURE AND OF THE ELECTRICPOLARIZATION WITH FIELD ALIGNED PARALLEL TO B 165

cycloidal structure in the ac plane, perpendicular to the external field, as expectedfor an antiferromagnetic material above the spin-flop transition. The treatmentof the data recorded at 5 T, i. e. in an intermediate field, revealed that the mag-netic structure is such that the rotation plane of the moments is in between theinitial and final orientations, and hence, suggests that the transition occurs in acontinuous manner. Taking into account the refinement at 5 T, the plane where themoments rotate tilts around the easy axis; one can observe in the Table 11.1 that byfixing the easy axis and leaving the rest of the parameters free for the refinement,the plane rotates around α towards the ac plane making an angle of κ = 32(4)˝with the ac plane. However, if we give more freedom to the structure, and refinethe parameters associated to α too, there is a small reorientation of the easy axis, ittilt towards c. In this last model the angle κ is bigger, 46(4)˝.

Temperature T = 2 K T = 2 K T = 2 K T = 2 KMagnetic field (H}b) 0 T 5 T 5 T 10 TInstrument D23 D23 D23 D23k k0 kb kb kb

Mn1 �(m)(μB) 3.83(7) 3.88(3) 3.86(4) 3.91(6)φu 0˝ 0˝ 0˝ 0˝θu 77(2)˝ 77˝ 64(4)˝ 86(2)˝�(m)(μB) -3.89(6) -3.43(4) -3.35(4) -3.30(4)φv 90˝ 106(3)˝ 124(4)˝ 180˝θv 90˝ 32(3)˝ 46(4)˝ -4(2)˝

Mn2 u2 = - u1

v2 = - v1

Δϕ kz2

kz2

kz2

kz2

κ 90˝ 32(4)˝ 37(4)˝ 0˝

RF2 11.8 12.3 11.5 14.5RF2w 10.0 9.58 8.71 13.1RF 16.1 17.2 16.8 19.0χ2 10.2 8.40 7.00 4.01

TABLE 11.1: Parameters that describe the magnetic structures of the Mn0.95Co0.05WO4 at2 K at zero field, 5 T and 10 T (H ‖ b). For simplicity the following notation has been used:k0 = [-0.216(1), 1

2 , 0.461(1)] and kb = [-0.216(2), 12 , 0.460(2)]. The parameter κ is the angle

between the plane where moments lay and the ac plane; it is not refined but derived fromφv and θv.

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166 CHAPTER 11. MAGNETIC-FIELD-INDUCED TRANSITIONS IN MN0.95CO0.05WO4

(a) T = 2 K H = 0 T

(b) T = 2 K H = 5 T

(c) T = 2 K H = 5 T

(d) T = 2 K H = 10 T

0 2 4 60

2

4

6

RF2w = 9.58 %RF = 17.2 %

T = 2 KH = 5 Tα fixed

(Fca

lc)2 (a

rb. u

nits

)

(Fobs)2 (arb. units)

0 2 4 60

2

4

6

RF2w = 13.1 %RF = 19.0 %

(Fca

lc)2 (a

rb. u

nits

)

(Fobs)2 (arb. units)

T = 2 KH = 10 TAF2’

H // ba

c

ba

c

0 2 4 6 80

2

4

6

8

RF2w = 10.0 %RF = 16.1 %

T = 2 KH = 0 TAF2

(Fca

lc)2 (a

rb. u

nits

)

(Fobs)2 (arb. units)

a

b

c

𝜶 ⟲

b

𝜶

𝜶

𝜅 = 32(4)o

0 2 4 60

2

4

6

RF2w = 8.71 %RF = 16.8 %

T = 2 KH = 5 T

(Fca

lc)2 (a

rb. u

nits

)

(Fobs)2 (arb. units)

H // ba

c

⟲𝜶’

b

𝜶’𝜅 = 37(4)o

H // b

a

c

ac plane

ac plane

FIGURE 11.5: Sketch of the magnetic structures of Mn0.95Co0.05WO4 at 2 K, only theellipse that envelops the moments is given: (a) projection along b and c axes at zero field;(b) and (c) projections along b and α under 5 T field, the latter refining the direction ofα and (d) projection along b of the high-field phase (10 T). κ is the angle of the rotationplane with respect to the ac plane.

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11.2. CONTINUOUS ROTATION OF THE MAGNETIC STRUCTURE AND OF THE ELECTRICPOLARIZATION WITH FIELD ALIGNED PARALLEL TO B 167

b

𝜶𝜅 = 108(3)o

b

𝜶𝜅 = 122(3)o

b

𝜶

𝜅 = 180o

b

𝜶𝜅 = 90o

T = 6.5 KH = 0 T

T = 6.5 KH = 3.5 T

T = 6.5 KH = 4 T

T = 6.5 KH = 6.2 T

(a) (b) (c) (d)

ac plane ac plane ac plane ac plane

0 2 4 60

2

4

6

RF2w = 7.82 %RF = 14.2 %

T = 6.5 KH = 0 TAF2

(Fca

lc)2 (a

rb. u

nits

)

(Fobs)2 (arb. units)

0 2 4 6 80

2

4

6

8

RF2w = 8.72 %RF = 18.5 %

T = 6.5 KH = 3.5 T

(Fca

lc)2 (a

rb. u

nits

)

(Fobs)2 (arb. units)

0 2 4 6 80

2

4

6

8

RF2w = 8.59 %RF = 18.1 %

T = 6.5 KH = 4 T

(Fca

lc)2 (a

rb. u

nits

)(Fobs)

2 (arb. units)

0 2 4 6 80

2

4

6

8

(Fca

lc)2 (a

rb. u

nits

)

(Fobs)2 (arb. units)

T = 6.5 KH = 6.2 T

RF2w = 9.40 %RF = 17.8 %

FIGURE 11.6: Evolution of the rotation plane of the moments with increasing the mag-netic field applied along b.

Temperature T = 6.5 K T = 6.5 K T = 6.5 K T = 6.5 KMagnetic field (H}b) 0 T 3.5 T 4 T 6.2 TInstrument 6T2 6T2 6T2 6T2k k0 kb kb kb

Mn1 �(m)(μB) 4.74(5) 4.87(4) 4.85(5) 4.86(3)φu 0˝ 0˝ 0˝ 0˝θu 70(2)˝ 69(2)˝ 65(2)˝ 66(2)˝�(m)(μB) -3.56(5) -3.59(4) -3.60(5) -3.62(3)φv 90˝ 80(2)˝ 73(2)˝ 0˝θv 90˝ 106(2)˝ 118(2)˝ 156(2)˝

Mn2 u2 = - u1

v2 = - v1

Δϕ kz2

kz2

kz2

kz2

κ 90˝ 108(3)˝ 122(3)˝ 180˝ = 0˝

RF2/% 10.2 12.3 11.6 11.8RF2w/% 7.82 8.72 8.59 9.40RF/% 14.2 18.5 18.1 17.8χ2 2.83 1.38 1.36 1.58

TABLE 11.2: Parameters that describe the magnetic structures of the Mn0.95Co0.05WO4 at6.5 K under 0 T, 3.5 T, 4 T and 6.2 T field (H ‖ b). For simplicity the following notationhas been used: k0 = [-0.216(1), 1

2 , 0.461(1)], kb = [-0.216(2), 12 , 0.460(2)]. The parameter

κ is not refined but deduced from φv and θv, and corresponds to the angle between thespin-rotation plane and the ac plane.

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168 CHAPTER 11. MAGNETIC-FIELD-INDUCED TRANSITIONS IN MN0.95CO0.05WO4

According to the experiment performed on D23, the transition occurs con-tinuously by the rotation around the easy axis of the plane where spins lay. Therotation of the plane was better studied at 6.5 K on 6T2 in another experiment, seeTable 11.2. In that experiment four data collections were done: zero field, 3.5 T, 4 Tand 6.2 T. The refinements of the data clearly conclude that indeed the plane ro-tates from the αb plane to the ac plane around α. Note that the rotation takes placein the opposite direction compared to the experiment performed on D23. This maybe because in one experiment the field was probably along b whereas in the otherit was along -b. The amplitudes of the ordered magnetic moments barely changealong the transition, and therefore, neither does the eccentricity.

If one simulates the evolution of the magnetic peaks in Fig. 11.4 with the pro-posed model for the transition, i. e. by considering that the plane rotates aroundα, the pattern that follow those magnetic reflections is reproduced rather well.Figure 11.7 presents the experimental integrated intensities of the (0 0 0)+k and(0 0 1)´k reflections (represented by empty symbols) as a function of the externalfield and the calculated intensities (full symbols) for those reflections as a functionof κ. There is an offset between the field and κ, however the way how the transitionoccurs is relatively well reproduced. Note that the second order effects revealed bythe refinements, such us the small re-orientation of the easy axis or the variation ofthe amplitudes, were not taken into account in the simulation.

80 60 40 20 0

(000)+kcalc

(000)+kobs

(001)-kcalc

(001)-kobs

Inte

nsity

(arb

. uni

ts)

κ (o)

2.0

2.5

4 5 6 7μ0H (T)

90

100

Inte

nsity

(arb

. uni

ts)

FIGURE 11.7: Observed integrated intensities of the (0 0 0)+k and (0 0 1)´k reflectionas a function of the field along b represented by (˛) and (˝) respectively. The simulatedintensities of those two peaks as a function of κ have (˛) and (‚) symbols.

The ferromagnetic component that may appear at high fields was also inves-tigated. A bunch of nuclear peaks were recorded under the different fields, they

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11.2. CONTINUOUS ROTATION OF THE MAGNETIC STRUCTURE AND OF THE ELECTRICPOLARIZATION WITH FIELD ALIGNED PARALLEL TO B 169

were carefully compared, but none of them showed any observable change whichcould be safely attributed to a ferromagnetic component. Moreover, the structuredoes not suffer any distortion, within experimental resolution.

11.2.2 Polarization

Figures 11.8(a) and 11.8(b) show the field dependence of the a, b and c compo-nents of the electric polarization at different temperatures, in H ‖ b configuration.A smooth decrease of Pb is observed starting at about H „ 3.5 T and finishing atabout 7.5 T (2 K), in good agreement with the other techniques. As observed in thesusceptibility measurements, the width of the transition decreases with increas-ing temperature and this is also reflected in the polarization measurements (notethat at 10 K Pb disappears within 3 T-field range). Polarization measurements ex-hibit a small hysteresis with the field. a and c components of the polarizationarise at about 3 T and reach their maximum at about 6.5 T (4.2 K). This resultsagree with the continuous flop of the rotation plane of the moments. In the initialand final stages, the polarization is either along b or in the ac plane, whereas inthe intermediate stages the electric polarization has non-zero components alongthe 3 directions. One may think that this can be due to a phase separation andtherefore, the result of the contribution of two elliptical-magnetic structures eachof which producing polarization either along b or in the ac plane. However, themagnetic-diffraction data refinement concluded that there is a unique phase andthat hypothesis can be ruled out.

Electric polarization along the a axis was measured as a function of the tem-perature at fixed fields. The first conclusion is that the TFE does not feel the pres-ence of the external field, within the experimental resolution: it is unchanged atleast up to 5 T fields. The behavior of the polarization is rather peculiar: at lowfields, it has a maximum value close to TFE and decreases at low temperatures. Arecent work by Liang et. al. [129] associates this interesting response of Pa to thetemperature at fields between 2 T and 5 T to the rotation of the spin-plane aroundthe easy axis. According to Liang in that field range and right below TFE, the spinplane is parallel to ac, and then, with decreasing temperature the plane rotates.The temperature range where the reorientation of the plane takes place dependson the field; for example at 2 T, Pa tends to zero within 2 K. On the contrary, at 4 Tnone of the components of the polarization is zero and therefore, at that field, therotation of the spin plane is not completed.

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170 CHAPTER 11. MAGNETIC-FIELD-INDUCED TRANSITIONS IN MN0.95CO0.05WO4

0 2 4 6 8 10 12 14

0

20

40

60 5 T

4 T

3 T

Pa

(μC

/m2 )

Temperature (K)

H//b

μ0H = 2 T

(c)

0 5 10 15

0

10

20

30

40

Pc

T = 12 K

T = 4.2 K

Pa,

c (μC

/m2 )

μ0H (T)

H//b

Pc

Pa

(b)

T = 4.2 K

0 5 10 15

0

10

20

30

40Temperature (K)

2 6 8 10 14

P

b (μ

C/m

2 )

μ0H (T)

H//b(a)

FIGURE 11.8: Field dependence of the components of the electric polarization for H}b,(a) Pb component at several temperatures and (b) Pa and Pc components at 4.2 K.(c) Temperature dependence of Pa at different fields.

11.3 Magnetic transitions with H ‖ α

Ac susceptibility measurements suggest a beginning of field-induced transitionwith magnetic field along the easy axis (see Fig. 11.1) that is probably completedslightly above the maximum available field for this measurements (9 T). The usualprocedure was followed to perform the experiment on D23 and first, two magneticreflections were monitored at 2 K as the field increased, from 0 T to 12 T. The inte-grated intensity of those two reflections display a clear and sharp transition at 10T, see Fig. 11.9. The intensity of the (0 0 -1)+k reflection gradually decreases untilthe transition occurs, then, it remains constant up to 12 T field. On the contrary,the intensity of the (-1 1 1)´k reflection increases and has a maximum value at 10T, then it decreases and above 11 T it becomes constant too.

Magnetic data were collected at 8 T (before the transition), 10 T and 11.75 T(above the transition). Unfortunately, the collected data at both high field and in-termediate fields did not allow us to determine unambiguously the correspondingmagnetic phases. Most likely, due to the geometrical restrictions imposed by themagnet, the set of collected reflections was not enough to give a definitive answer,and therefore this experiment should be repeated.

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11.4. SUMMARY AND CONCLUSIONS 171

0 2 4 6 8 10 12

1.3

1.4

1.5

(-111)-k

(00-1)+k

(Fob

s)2 (arb

. uni

ts)

μ0H (T)

T ~ 2 KH//

5

6

7

(Fob

s)2 (arb

. uni

ts)

FIGURE 11.9: Integrated intensities of (0 0 -1)+k and (-1 1 1)´k reflections as a function ofthe external field applied along α at 2 K.

11.4 Summary and conclusions

To summarize, the effect of a magnetic field applied along the b axis ofMn0.95Co0.05WO4 has been investigated and this study consistently reveals that,the ferroelectric phase with spin-cycloid structure in the αb plane and with polar-ization along b undergoes a continuous transition from the αb plane to the wα plane,with w = α + 90˝ being the hard direction. Such evolution of the plane rotationwas suggested by Liang et. al. [129] but only our neutron diffraction study unam-biguously confirmed it. The electric polarization undergoes a similar behavior: itrotates continuously from the b axis to the ac plane. Indeed, as expected from therotation of the spin-plane. As a result, we propose a complete phase diagram thatgathers all the results we obtained, see Fig. 11.10.

Although both, bulk magnetic measurements and the evolution of certainmagnetic reflections clearly demonstrate a field-induced transition, with H ‖ α,neutron-diffraction experiments were not conclusive about the spin arrangement.

By comparing the H ´ T phase diagrams with H ‖ b of the MnWO4 andMn0.95Co0.05WO4 compositions it becomes evident that doping MnWO4 with only5% of cobalt does not only to destabilize the AF1 phase. The influence of an ex-ternal magnetic field along the monoclinic-b axis in both compounds is radicallydifferent. In the parent compound the mb component of the magnetic moment de-creases with increasing field until it becomes unstable and flops to the ac plane.This is not the case in Mn0.95Co0.05WO4, where the rotation happens in a continu-

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172 CHAPTER 11. MAGNETIC-FIELD-INDUCED TRANSITIONS IN MN0.95CO0.05WO4

0 2 4 6 8 10 12 140

2

4

6

8

10 AF2’Pa + Pc

AF3

μ0H

(T)

Temperature (K)

Par

amag

netic

AF2Pb

Pa + Pb + Pc

FIGURE 11.10: H ‖ b phase diagram proposed for Mn0.95Co0.05WO4.

ous manner. One main consequence is that in the MnWO4, Pb and Pa do not coexistwhereas in the doped composition they do.

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Chapter 12Magnetic-field-induced

transitions in Mn0.90Co0.10WO4

Various magnetic-field-induced phase transitions will be presented in thischapter, which were studied by means of magnetic and magnetoelectricmeasurements, and neutron diffraction.

Two magnetic structures appear below TN in Mn0.90Co0.10WO4: AF3 andAF2’. The AF3 is described as a cycloidal structure with the magnetic momentslocated basically within the basal plane of the octahedra. The AF2’ phase, the mul-tiferroic one, corresponds to a cycloidal order within the ac plane. The magneticsymmetry of the latter structure is different to that of the multiferroic phases ofthe rest of the studied compositions. So, applying a magnetic field along differentdirections could give rise to effects that may only happen to this composition.

Please note, that this was the first under-field-study done in this thesis. Wemade the mistake of applying the field along the crystallographic axes instead ofapplying it along the magnetically characteristic ones. Nevertheless, the easy axisis 9˝ from a at 2 K, and consequently the field has been applied rather close to themagnetic-easy axis.

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174 CHAPTER 12. MAGNETIC-FIELD-INDUCED TRANSITIONS IN MN0.90CO0.10WO4

12.1 Bulk magnetic response

The field dependence of the magnetization at 2 K along the principle crystallo-graphic directions is shown on Fig. 12.1. Up to fields as high as 9 T the magne-tization along b and c axis reminds higher than along the a axis. While the laterchanges linearly up to about 7.5 T, it turns concave upwards at higher fields sig-nalling a spin reorientation process, presumably of spin-flop type. More carefulinspection of the c axis magnetization reveals similar but less pronounced pecu-liarity at fields of about 2 T, see the calculated derivative of the magnetization inthe insert of Fig. 12.1. No coercivity was observed in any orientation.

0 2 4 6 8 100

5

10

15

Mag

netiz

atio

n (e

mu/

g)

μ0H (T)

T = 2 K

bc

a

0 2 4 61.35

1.50

1.65

μ0H (T)

δMc/δ

(μoH

) (em

u/gO

e)

FIGURE 12.1: Magnetization curves along the principle crystallographic directions. Inthe insert, the derivative of the c-axis magnetization with respect to the applied field.

To better monitor the above field-induced departure from the linear variationof magnetization with field applied along a and c directions, we studied the ac-susceptibility at fixed temperatures as a function of the amplitude of the dc-field,superimposed in the same direction as the ac-field. As illustrated on Fig. 12.2(a), at2 K the field-induced transition along the a direction starts at fields as low as 6.5 Tas evidenced by the increase of the real component of the ac-susceptibility, whichforms a peak at 8.7 T. This critical field should correspond to the maximum slopein the magnetization curve depicted on Fig. 12.1. As suggested by Fig. 12.2(a), thefield-induced transition is not completed up to the highest applied field of 9 T. Thecritical field slightly decreases with increasing temperature (to about 8.3 T at 10 K).A small but noticeable hysteresis is seen depending on the way the field is ramped,see the inset of Fig. 12.2(a). No clear maximum of ac-susceptibility is observedbetween TN2 and TN . However, one should keep in mind that this narrow temper-

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12.1. BULK MAGNETIC RESPONSE 175

0 2 4 6 8 10

1.5

2.0

2.5

3.0

μ0H (T)

a axis

hac

= 10 Oe

15 12

8.5 7.5 5

2

T(K)

χ’ (1

0-4 C

GS

)

(a)

8.0 8.4 8.8

2.0

2.5

3.0 T = 2 K

0 2 4 6 81.3

1.4

1.5

1.6

1.7

13

12

10

6 8

2

χ’ (1

0-4 C

GS

)

T(K)

c axishac = 10 Oe

(b)

0 1 2 3

1.4

1.6

T = 2 K

FIGURE 12.2: (a) Real component of ac-susceptibility along a-axis as a function of super-imposed dc field amplitude. (b) The same for the c axis. The insets are zooms of thesusceptibility maxima. The arrows indicate the direction of changing the field.

ature region is too close to the paramagnetic transition, which might be affectedby the relatively high magnetic field. Figure 12.2(b) focuses on the hardly seenmagnetization anomaly along c direction, depicted on Fig. 12.1, and demonstratesthe versatility of the used approach in studying small changes in the magnetiza-tion curve. At 2 K a clear maximum of the ac-susceptibility is observed at about2.4 T (as also manifested on from the derivative of the magnetization), suggestinga field-induced transition. Such transition was not reported in the previous studies[126]. The width of the peak implies that this transition is accomplished in a rela-

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176 CHAPTER 12. MAGNETIC-FIELD-INDUCED TRANSITIONS IN MN0.90CO0.10WO4

tively broad range of fields between about 1 to 3.5 T. A very narrow but noticeablehysteresis is visible on the inset of 12.2(b). The peak position is strongly affectedby temperature and it shifts to 3.2 T at 10 K. Importantly the susceptibility peakdisappears above TN2.

0 2 4 6 8 10300

400

500

600

(0 1 0)+ k

T = 2 K H // a

(Fob

s)2 (arb

. uni

ts)

μ0H (T)

(b)

0 1 2 3 4 5750

825

900

(0 0 1)-k

T = 2 K H // c

(Fob

s)2 (arb

. uni

ts)

μ0H (T)

(a)

FIGURE 12.3: (a) Evolution of the integrated intensity of (0 0 1)´k magnetic satellite withthe magnetic field applied parallel to the c axis (H}c). (b) Similar plot for the (0 1 0)+k

satellite when the magnetic field is applied parallel to the a axis (H}a).

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12.2. NEUTRON DIFFRACTION 177

12.2 Neutron diffraction

12.2.1 Field along the c axis

In an experiment performed on the diffractometer D15, where the sample wasmounted with the c axis parallel to the field we have followed the (0 0 1)´k mag-netic satellite as a function of the applied field at T = 2 K. The variation of its inte-grated intensity is depicted in Fig. 12.3(a). The field-induced transition suggestedby the magnetization and ac-susceptibility measurements with H}c is apparent.The transition starts at fields around 2 T and is completed at about 3 T, in agree-ment with those measurements. Data were then collected at 5 T. First, the prop-agation vector was refined by centering 35 magnetic peaks which revealed that it isnearly unchanged compared to the zero field value: k(H =5T) = [-0.220(1), 1

2 , 0.469(2)].

Temperature T = 2 K T = 2 K T = 2 K T = 2 KMagnetic field 0 T 2.2 T } c 5 T } c 11 T } ak k0 kc kc ka

Mn1 �(m)(μB) 4.45(2) 3.21(8) 4.41(2) 3.67(3)φu 0˝ 0˝ -7(2)˝ 90˝θu 99(2)˝ 87(3)˝ 95(1)˝ -21(2)˝�(m)(μB) 4.03(2) 4.10(8) 3.78(2) 4.19(3)φv 0˝ 0˝ 83(2)˝ 83(1)˝θv 9(2)˝ -3(3)˝ 93(1)˝ 69(2)˝

Mn2 u2 = - u1 u2 = - u1 u2 = - u1 u2 = - u1

v2 = - v1 v2 = - v1 v2 = - v1 v2 = - v1

Δϕ kz2

kz2

kz2

kz2

RF2/% 5.67 9.89 5.81 9.53RF2w/% 6.16 10.5 5.86 7.33RF/% 6.65 11.9 5.39 17.0χ2 7.14 32.1 8.37 8.30

TABLE 12.1: Parameters that describe the magnetic structures of the Mn0.90Co0.10WO4

at 2 K under 5 T field along the c axis and 11 T field along the a axis. Zero fieldstructure from Chapter 8 is also included for comparison. For simplicity the follow-ing notation has been used: k0 = [-0.222(1), 1

2 , 0.472(1)], kc = [-0.220(1), 12 , 0.469(2)] and

ka = [-0.206(3), 12 , 0.465(1)].

The integrated intensities of 135 independent magnetic peaks were collectedand refined first at 2.2 T and then at 5 T. The obtained parameters are gathered

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178 CHAPTER 12. MAGNETIC-FIELD-INDUCED TRANSITIONS IN MN0.90CO0.10WO4

in Table 12.1 and the corresponding structures are depicted in Figs. 12.4(a) and12.4(b). At 2.2 T, the data can be explained with relatively good agreement as a cy-cloidal structure with the spins in the ac plane. This structure keeps the symmetryof the zero field AF2’ phase, with the small differences such as a small change in theorientation of the principal axes of the ellipse and the amplitude of the momentscloser to a and an eccentricity ε(H}c = 2.2 T) = 0.62 (the direction perpendicularto the field). At higher fields, the scenario is completely different. Compared tothe zero-field structure, there is a flip of the plane where the moments rotate fromthe ac plane towards the ab plane. The refinements improve if we consider that therotation plane is not parallel to the ab plane but slightly tilted, as the parameters inTable 12.1 show. That plane is forced by the field to be as perpendicular as possibleto the direction of the external field. In addition, the eccentricity of the ellipse inthe new structure has increased, from ε(H = 0) = 0.42 to ε(H}c = 5 T) = 0.51.

0.0 0.5 1.00.0

0.5

1.0

RF2w = 5.86 %RF = 5.39 %

(Fca

lc)2 (a

rb. u

nits

)

(Fobs)2 (arb. units)

Rotation plane: (ab)H // cT = 2 K

(a)

(b)

(c)c

a

b

a

FIGURE 12.4: Magnetic structure of Mn0.90Co0.10WO4 determined at 2 K and 5 T withthe magnetic field applied parallel to the c axis (H}c). (a) A projection in the ac plane ofthe spin order. (b) Projection of the magnetic structure in the ab plane. The spin rotationplane is shown in light purple. (c) Agreement plot of the magnetic refinement.

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12.2. NEUTRON DIFFRACTION 179

The influence of the magnetic field on the AF3 phase was also explored. Neu-tron diffraction data were collected at 12 K and 5 T. Under these conditions themagnetic structure remains collinear and sinusoidally modulated with the samepropagation vector. Only a deviation of the moment direction with respect to thec˚ axis was observed. The moments remain in the ac plane, but the angle θ formedwith the c˚ axis decreases: 112(1)˝ instead of the 124.8(5)˝ at zero field. Hence, theeffect of the field along the c axis on AF3 is to rotate the spins towards the ab plane,in order to diminish their projection onto the direction of the external magneticfield. Thus the field moves the spins away from the equatorial plane of the MO6

octahedra. This is in good agreement with the ac-susceptibility, from which oneshould not expect big changes of the AF3 magnetic structure.

Regarding the nuclear structure, the external field does not alter the crystalstructure within experimental resolution, and no ferromagnetic component wasobserved either.

12.2.2 Field along the a axis

As shown in Figs. 12.1 and 12.2, a magnetic field applied along the a axis alsoinduces a magnetic transition in the AF2’ structure. In this case, higher field isneeded to achieve the spin reorientation and, in addition, the transition is muchwider than the one along the c axis. The field dependence of the magnetic satellite(0 1 0)+k, depicted in Fig. 12.3(b), shows that at 2 K the transition takes placebetween 5.5 T and 9.5 T. Above 9.5 T the magnetic transition is completed. Withdecreasing field the integrated intensity exhibits hysteresis. Analyzing the data,a complex behavior of the reflection was observed. Scans along ω angle of thediffractometer were done at increasing and decreasing field, see Figs. 12.5(a) and12.5(b). With increasing field, the magnetic reflection splits in two at about 7.3 T tofinally collapse in a unique peak at about 9.3 T. The resulting reflection is displacedwith respect to the initial one which suggests a modification of the propagationvector k. With decreasing field, the splitting of the peak occurs at little lower fields,at about 9 T, at they join back at about 6 T. After undergoing the field ramp, thehigh field propagation vector becomes stable at zero applied field.

We have studied this new spin order by collecting neutron data at 11 T and2 K on the D23 diffractometer. Centering of several magnetic reflections concludedthat the propagation vector slightly changes compared to the one at zero field be-coming k = [-0.206(3), 1

2 , 0.465(1)]. The spins arrangement was determined based

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180 CHAPTER 12. MAGNETIC-FIELD-INDUCED TRANSITIONS IN MN0.90CO0.10WO4

-45 -44 -43 -42 -410

2

4

6

8

10

(000)+k

Decreasing field

μ0H

(T)

ω (o)

40.00

1145

2250

-45 -44 -43 -42 -410

2

4

6

8

10

(000)+k

Increasing fieldμ

0H(T

)

ω (o)

FIGURE 12.5: ω scan of (010)+k reflection followed at 2 K as (a) the applied field increasedand (b) decreased.

on the collected 125 independent reflections. The magnetic structure at 2 K and11 T is shown in Fig. 12.6 and is described in Table 12.1. The main effect of the fieldalong a is to flip the plane of the spin cycloid from the ac plane (H = 0) towardsthe bc plane (H}a = 11 T). Moreover, as seen in Table 12.1, we observe a significantdecrease in the amplitude of the ordered moments. In the H}a configuration, theeccentricity (ε = 0.48) is slightly larger compared to the ground state.

The evolution of the AF3 phase with the magnetic field was not studied sincefrom the macroscopic measurements we did not expect any difference but tiltingof the moments towards the bc plane, in analogy with field along the c axis. Nei-ther crystal-structure changes nor ferromagnetic component were detected at 11 K,within the experimental resolution.

12.3 Polarization

Figure 12.7(a) shows the dependence of the a, b and c components of the polariza-tion as a function of the magnetic field applied along the c axis at T = 2 K. A sharpdecrease of both Pc and Pa components is observed beginning at a threshold fieldHc „ 2 T. With increasing the temperature up to TN2 „ 11 K, the evolution of thePc (Hc) and Pa(Hc) remains the same with only the initial polarization decreasingand the threshold field increasing. The temperature dependence of the thresh-old field for field-induced transition (see the insert in Fig. 12.7) coincides withthe one found from the magnetization and ac susceptibility curves along c axis(Figs. 12.1 and 12.2). The inverse Dzyaloshiskii-Moriya interaction in the chains

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12.3. POLARIZATION 181

0 2 40

2

4Rotation plane: (bc)H // aT = 2 K

RF2w = 7.33 %RF = 17.0 %

(Fca

lc)2 (a

rb. u

nits

)

(Fobs)2 (arb. units)

(a)

(b)

(c)c

a

c

b

FIGURE 12.6: Magnetic structure of Mn0.90Co0.10WO4 determined at 2K and 11 T with themagnetic field applied along a (H}a). Projection of the magnetic structure in the ac plane(a) and in the bc plane (b). (c) Calculated structure factors are plotted against the experi-mental ones.

along a axis could contribute to the Pb component of the electric polarization inthis field-induced phase. However, only a small variation of Pb component wasobserved at the threshold field which became more noticeable with increasing T(Fig. 12.7). Whether the small Pb polarization is related to the weaker magneto-electric parameter along b axis or to a formation of the ferroelectric domains inthe field-induced phase (containing of both right- and left-handed spin-cycloids)remains unclear and requires further study.

When the magnetic field is applied parallel to the a axis, similar suppressionof the Pa polarization was observed at pulsed field „ 8.5 T [Fig. 12.7(b)]. Thisis in agreement with the non-linear increase of magnetization (Fig. 12.1) and themaximum in the field dependence of the ac susceptibility [Fig. 12.2(a)]. It indicatesa field-induced spin-flop transition of the cycloid to the bc plane perpendicular tothe field.

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182 CHAPTER 12. MAGNETIC-FIELD-INDUCED TRANSITIONS IN MN0.90CO0.10WO4

0 5 10 15 200

25

50

75

H // c

H // b

H // a

T = 4.2 K

Pa (

μC

/m2 )

μ0H (T)

(b)

0 1 2 3 4 50

25

50

75

100

Pb (5 K)(10K)

Pc (2 K)

Pa (2 K)

P (μC

/m2 )

μ0Hc (T)

0 5 10 150

10

20

30

TN2 T

N

Hcr (k

Oe)

T (K)

(a)

FIGURE 12.7: (a) Dependence of electric polarization along a, b and c axes under magneticfield along the c-axis. Insert shows the evolution of the critical field as a function of thetemperature, Hcr-T diagram, obtained from electric polarization and magnetization data.(b) Pa component of the electric polarization in pulsed magnetic field along a-, b- andc-axes, exhibiting a suppression of Pa for H}a and H}c due to the flop of the plane of thecycloidal structure perpendicular to the field. Almost nothing changes for H}b, since b isalready orthogonal to the rotational plane.

No substantial effects for the electric polarization was found in pulsed mag-netic field up to 15 T applied along the b axis confirming the absence of field-induced transitions in this direction, in agreement with the magnetic data in Fig. 12.1and with Ref. [125].

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12.4. SUMMARY AND CONCLUSIONS 183

12.4 Summary and conclusions

To summarize, magnetic-field-induced phase-transitions in Mn0.90Co0.10WO4 mul-tiferroic were explored using magnetic and magnetoelectric measurements, andneutron diffraction. Our data consistently reveal that, the multiferroic AF2’ phaseremains stable in magnetic field as high as 15 T applied along the b axis, whilein field along c and a axes the spin cycloid structure flops from the ac plane todirections almost perpendicular to the magnetic field at „ 3 T and „ 8.5 T, re-spectively. Those spin reorientations are accompanied by a dramatic suppressionof both Pa and Pc components of the polarization. The refinement of the neutrondiffraction data has allowed us to extract the characteristic parameters includingwave vectors, orientations of the main elliptical axes of the magnetic structures inthe different spontaneous and field-induced phases.

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184 CHAPTER 12. MAGNETIC-FIELD-INDUCED TRANSITIONS IN MN0.90CO0.10WO4

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Chapter 13Magnetic-field-induced

transitions in Mn0.80Co0.20WO4

The evolution under applied magnetic field of the magnetic structures of thex = 0.20 composition of the Mn1´xCoxWO4 family has been investigatedin this chapter.

13.1 Magnetic and ferroelectric properties under fieldsperpendicular to α

13.1.1 Re-orientation of the conical antiferromagnetic struc-ture with H ‖ b

Figure 13.1(a) shows the temperature dependence of the magnetic ac-susceptibilityat fixed dc-fields. One can see that the Néel temperature does not vary up to 9 T.Down to the second transition („ 9 K) all the curves collapse in the same line, butbelow this temperature, the evolution of the susceptibility depends on the inten-sity of the external field. As the field increases, the susceptibility increases until itreaches a maximum. The maximum has a higher value at higher fields, and thatmaximum is reached at lower temperatures. After the transition, all the curvesdecrease in a similar manner.

The behavior of the electric polarization along b in the ferroelectric phase of

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186 CHAPTER 13. MAGNETIC-FIELD-INDUCED TRANSITIONS IN MN0.80CO0.20WO4

Mn0.80Co0.20WO4 with an external field applied along b is depicted in Fig. 13.1(b).With increasing field, Pb decreases and disappears rather abruptly at about 11 T.This transition shows a large hysteresis when ramping down the field.

According to bulk magnetic and polarization measurements, the conical struc-ture is not stable under field aligned parallel to b and tends to disappear at highfields. On the contrary the AF4 phase remains stable, and seems to extend its sta-bilization range to low temperature at the expenses of the conical structure. How-ever, the latter point has to be taken with some precaution, since we are dealingwith partial information, χb and Pb.

0 5 10 15 200

10

20

30

40

Pb (μ

C/m

2 )

μ0H (T)

T = 4.2 KH // b

(b)

5 10 15 20 25 30

1.15

1.20

1.25

1.30

μ0H (T) 0 T 7 T 2 T 8 T 4 T 9 T 6 T

Temperature (K)

χ’ (1

0-4 C

GS

)

H // bhac = 10 Oef = 1111 Hz

(a)

FIGURE 13.1: (a) Temperature dependence of the real ac-susceptibility at fixed superim-posed dc fields along b. (b) Field dependence of the component along b of the electricpolarization at 4.2 K.

With the aim of establishing the magnetic structures within the (H ‖ b) ´ T

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13.1. MAGNETIC AND FERROELECTRIC PROPERTIES UNDER FIELDS PERPENDICULAR TO α 187

4 6 8 10 12 14 16 18 20 220

50

100

150

200

250μ0H//b = 12 T

(0 0 0)+k1

(1 -1 -1)+k2

TN2

Inte

nsity

(arb

. uni

ts)

Temperature (K)

TN

(b)

0 2 4 6 8 10 12

220

225

230

235 (0 0 0)+k1

Inte

nsity

(arb

. uni

ts)

μ0H (T)

H//bT = 6.5 K

20

30

40

(1 -1 -1)+k2

Inte

nsity

(arb

. uni

ts)

(a)

FIGURE 13.2: (a) Field dependence of the (0 0 0)+k1 and (1 -1 -1)+k2 magnetic reflectionsat 6.5 K with magnetic field along b. (b) Temperature dependence of the (0 0 0)+k1 and(1 -1 -1)+k2 reflections with 12 T field applied along b.

phase diagram, a neutron diffraction experiment was performed on D23.Figure 13.2(a) illustrates the field dependence of the (0 0 0)+k1 and (1 -1 -1)+k2 mag-netic reflections at 6.5 K with magnetic field along b, k1 = ( 1

2 , 0, 0) being the propa-gation vector that corresponds to the collinear AF4 phase andk2 « (-0.211, 1

2 , 0.452) the one associated to the incommensurate part of the con-ical structure. The evolution of the commensurate reflection does not show anyevident indication of any transition, in contrast to the behavior of the incommen-surate reflections. The intensity of this latter reflection has a downward tendencyup to 8 T: above this field, the intensity becomes constant. It is worth noting thatthe intensity of the incommensurate peak, even weak, is non-zero above the tran-sition, and consequently, the hypothesis that the AF4 phase alone was extended tolower temperatures is ruled out immediately. The existence of the incommensurate

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188 CHAPTER 13. MAGNETIC-FIELD-INDUCED TRANSITIONS IN MN0.80CO0.20WO4

wave vector was double-checked by monitoring commensurate and incommensu-rate reflections, as the temperature increases under high field [Fig. 13.2(b)]. Thefigure shows that the incommensurate reflection exists up to around 8.7 K, coincid-ing with the TN2 transition temperature attributed to the AF4-to-conical transitionat zero field, see Chapter 8.

According to these observations, the new high-field state must correspond toa double-k structure. The integrated intensity of the magnetic satellites of (h k l)+k1

and (h k l)+k2 were collected at 6.5 K at zero field, 7 T and 12 T (about 45 reflectionsof each kind). The results of the refinements are gathered in Table 13.1 and confirmthat a conical spin-arrangement can explain the high field data. The collinear partof the conical structure remains unchanged whereas the incommensurate compo-nent, i. e. the cycloidal part, is modified. The plane where the magnetic momentsare located turns 90˝ and becomes perpendicular to the external field. The refine-ment of the data at 7 T suggest that the transition occurs as it does in the pureMnWO4 when the external field aligned parallel to b, that is, the mb-spin compo-nent decreases with increasing field until it flops perpendicular to the it.

Temperature T = 6.5 K T = 6.5 K T = 6.5 K T = 6.5 K T = 6.5 K T = 6.5 KMagnetic field (H ‖ b) 0 T 0 T 7 T 7 T 12 T 12 Tk k0

1 k02 k1 k2 k1 k2

Mn1 �(m)(μB) 2.47(2) 1.71(7) 2.49(2) 1.74(6) 2.47(2) 1.69(9)φu 0˝ 0˝ 0˝ 0˝ 0˝ 0˝θu 141(1)˝ 40(5)˝ 140(1)˝ 44(3)˝ 141(1)˝ 42(4)˝�(m)(μB) -1.70(2) -1.18(8) -0.6(1)φv 90˝ 90˝ 0˝θv 90˝ 90˝ -48(4)˝

Mn2 u2 = u1 u2 = - u1 u2 = u1 u2 = - u1 u2 = u1 u2 = - u1

v2 = v1 = 0 v2 = - v1 v2 = v1 = 0 v2 = - v1 v2 = v1 = 0 v2 = - v1

Δϕ 0 kz2 0 kz

2 0 kz2

RF2/% 10.3 14.0 9.25 12.5 8.99 12.6RF2w/% 5.40 10.6 5.12 8.11 4.79 6.85RF/% 15.3 20.1 13.5 16.6 13.1 26.8χ2 4.70 1.43 4.24 0.54 3.59 0.16

TABLE 13.1: Parameters that describe the magnetic structures of the Mn0.80Co0.20WO4 at6.5 K under 0 T, 7 T and 12 T field along the b axis. For simplicity the following notationhas been used: k1 = ( 1

2 , 0, 0) and k2 = [-0.215(1), 12 , 0.452(2)].

This new scenario with a high-field-conical structure explains the absenceof electric polarization along b. However, this structure could give rise to somepolarization within the ac plane, and therefore, this should be further studied.

Summarizing, applying an external magnetic field along the b axis of theMn0.80Co0.20WO4 compound only modifies the conical-antiferromagnetic struc-

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13.1. MAGNETIC AND FERROELECTRIC PROPERTIES UNDER FIELDS PERPENDICULAR TO α 189

ture by re-orienting the semi-axes of the cycloidal part of it. Concomitant to there-orientation of the rotation plane of the spins, the electric polarization along bdisappears, and probably flops onto the ac plane. As a result, we propose thephase diagram of Fig. 13.3 that takes into account all this information.

0 4 8 12 16 200

2

4

6

8

10

12

14

Collineal AF4 (P = 0)

Par

amag

netic

Hig

h fie

ld s

truct

ure

H//b

μ0H

(T)

Temperature (K)

Conical structure (Pb)

(Pa,Pc ?)

FIGURE 13.3: Phase diagram for Mn0.80Co0.20WO4 compound with H ‖ b. Circlescome from bulk magnetic measurements, triangles from polarization measurements andsquares from neutron diffraction.

13.1.2 Similar behavior for H ‖ α + 90˝

Bulk magnetic measurements and polarization measurements performed on thesame crystal with the field applied along the so-called hard-magnetic axis (α+ 90˝),revealed a behavior similar to the one observed with H ‖ b. The real component ofthe ac-susceptibility along α + 90˝ mimics the χ1

b as a function of the temperatureat different external fields. As observed in Fig. 13.4(a), the Néel temperature is thesame for fields up to 9 T, while the second transition (AF4-to-conical) is shifted tolower temperatures as the field increases. The transitions were observed only inthe conical phase and require more than 9 T below 5 K [see Fig. 13.4(b)].

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190 CHAPTER 13. MAGNETIC-FIELD-INDUCED TRANSITIONS IN MN0.80CO0.20WO4

The response of the b component of the electric polarization to the field par-allel to the hard magnetic axis also mimics the response of Pb(H ‖ b) to field. Itdisappears at about 12.5 T and exhibits a wide hysteresis (at 4.2 K), illustrated inFig. 13.4(c).

Therefore, based on these experimental evidences we propose a preliminaryphase diagram for this orientation, depicted in 13.4(d). The H ‖ α + 90˝ phasediagram is very much the same as the one for H ‖ b. Of course, further polariza-tion measurements a neutron diffraction experiments are necessary to confirm thishypothesis.

0 2 4 6 8 10 12 14 16 18 20 220

2

4

6

8

10

12

14 H// +90o

Conical structure Collinear AF4 (P = 0)

Hig

h fie

ld s

truct

ure

μ0H

(T)

Temperature (K)

Par

amag

netic

(d) Pa, Pc ?

0 5 10 15 200

10

20

30

40 T = 4.2 KH // +90o

Pb (μ

C/m

2 )

μ0H (T)

(c)

0 2 4 6 8

1.20

1.25

1.30

11 K5 K

H // + 90o

hac = 10 Oef = 1111 Hz

9 K 8 K 7.5 K

7 K

2 K

19 K

22 K

μ0H (T)

χ’ (1

0-4 C

GS

)(b)

5 10 15 20 25 301.10

1.15

1.20

1.25

1.30

μ0H (T) 0 T 7 T 3 T 8 T 5 T 9 T 6 T

Temperature (K)

χ’ (1

0-4 C

GS

)

H // +90o

hac = 10 Oef = 1111 Hz

(a)

FIGURE 13.4: (a) Temperature dependence of the real ac-susceptibility at fixed superim-posed dc fields along α + 90˝. (b) Field dependence of the real ac-susceptibility at fixedtemperatures with the field along α + 90˝. (c) Field dependence of Pb at 4.2 K. (d) H ´ Tphase diagram proposed for Mn0.80Co0.20WO4 compound with H ‖ α + 90˝ based onbulk magnetic measurements, polarization measurements and the results obtained withH ‖ b. Circles result from bulk magnetic measurements and triangles from polarizationmeasurements.

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13.2. STABILIZATION OF THE CYCLOIDAL AF2 PHASE IN H ‖ α CONFIGURATION 191

13.2 Stabilization of the cycloidal AF2 phase in H ‖ α

configuration

Applying a magnetic field along the easy-magnetic axis modifies completely thescenario. The stability of the magnetic structures is reversed compared to the pre-vious cases, i. e. the collinear structure is destabilized and disappears, and theincommensurate component of the cone replaces of the collinear part. Bulk mag-netic measurements [see Fig. 13.5(a)] show that the maximum in the susceptibility,attributed to the Néel temperature, moves to lower temperatures as the field is in-creased. Moreover, the AF4-to-conical transition, which is not visible in χα(H = 0)within the experimental resolution, is evidenced as the external field increases,and suggests a reorientation of the moments in the low temperature phase. As ob-served in Fig. 13.5(b) the behavior of isothermal χ in the conical structure suggestsa magnetic transition at field higher than 9 T. For the detection of those transi-tions, pulsed high fields were used and this technique demonstrates that the con-ical structure transforms into some other spin configuration at about 12.7 T field,see Fig. 13.5(c). The isothermal dM/dH curves reveal that the transition fromthe AF4 phase to presumably the paramagnetic state is broader than the transitionfrom the conical structure to the high field phase. Note that the curve labeled as9.3 K, has actually been measured at somewhat lower temperature which falls inthe region where the conical phase exists (below TN2 = 8.5 K). This discrepancyhas been attributed to the magnetocaloric effect (the change in the spin entropy asa function of magnetic fields) and the lack of time needed to reach thermal equi-librium during the short pulse duration (of the order of few msec). Though theinfluence of this effect on the temperature in our measurements has been firmlyestablished, it is not easy to say how much the real sample temperature differsfrom the reading of the thermometer (at least because the magnetocaloric effect atdifferent temperatures is different).

The response of the electric polarization along b, Pb, is completely differentto the previous cases. The Pb component of the electric polarization first increaseslinearly up to 12.5 T and then increases more rapidly, see Fig. 13.6. This behavioragrees with the previously proposed hypothesis: the field along α induces somemodification of the incommensurate part of the conical structure, such as a smallre-orientation of the semi axes or an increase of the mb component, and then atabout 12.5 T the collinear component of the structure disappears and contributesto the incommensurate one, increasing rapidly the Pb component.

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192 CHAPTER 13. MAGNETIC-FIELD-INDUCED TRANSITIONS IN MN0.80CO0.20WO4

0 10 20 300.5

1.0

1.5

2.0

2.5

3.0

1.5 K

7.5 K

9.3 K

14.9 K

17.2 K

μ0H (T)

dM/d

H (a

rb. u

nits

)

(c)

22 K

5 10 15 20 25 30

1.2

1.4

1.6

H // hac = 10 Oef = 1111 Hz

μ0H (T) 0 T 2 T 4 T 6 T 7 T 8 T 9 T

χ’ (1

0-4 C

GS

)

Temperature (K)

(a)

0 2 4 6 8 10

1.2

1.4

1.6

21 K

H // hac = 10 Oef = 1111 Hz

19 K

17 K

16 K

15 K

11 K

8 K

2 K

μ0H (T)

χ’ (1

0-4 C

GS

)

(b)

FIGURE 13.5: (a) Temperature dependence of the real ac-susceptibility at fixed superim-posed dc fields along α. (b) Field dependence of the real ac-susceptibility at fixed temper-atures with the field along α. (c) dM/dH isothermal curves obtained by pulsed magneticfields.

0 4 8 12 16 200

20

40

60

80 H//b H//c H// +90o H// H//a

Pb (μC

/m2 )

μ0H (T)

FIGURE 13.6: Field dependence of Pb at 4.2 K with H parallel to b, α, α + 90˝, a and c.

With all this information we have built phase diagram from the susceptibilityin H ‖ α (Fig. 13.7). The black rhombs represent the transition fields obtained bypulsed magnetic fields and because of the magnetocaloric effect they have movedto lower temperatures to match the ac-susceptibility data. The proposed AF2 struc-ture for the high field phase must be checked by neutron diffraction.

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13.2. STABILIZATION OF THE CYCLOIDAL AF2 PHASE IN H ‖ α CONFIGURATION 193

The effect of the field on the electric polarization was not only studied alongthe b axis and the main magnetic axes but also along the crystallographic ones.Figure 13.6 shows that Pb has a similar behavior when the field is along b, c orα + 90˝, and that this behavior is different when the field is along a or α. There-fore, it is reasonable to study by means of neutrons the effect of the field appliedalong a to try to elucidate which could be the high field phase in such case. For thatexperiment on D23, 6 T was the highest field available. However it was enoughto see that the amplitude of the commensurate part at 2 K slightly decreases com-pared to the zero-field AF4 structure (Table 8.4), whereas in the incommensuratepart there is a re-orientation of one semi axis towards the c axis (perpendicular tothe field) and an augmentation of the mb spin component. The refined parametersare summarized in Table 13.2.

Temperature T = 2 K T = 2 KMagnetic field (H ‖ a) 6 T 6 Tk k1 k2

Mn1 �(m)(μB) 3.04(5) 2.80(2)φu 0˝ 0˝θu 142(2)˝ 40(7)˝�(m)(μB) -3.0(1)φv 90˝θv 90˝

Mn2 u2 = u1 u2 = - u1

u2 = u1 v2 = - v1

Δϕ 0 kz2

RF2/% 5.67 6.41RF2w/% 6.16 6.30RF/% 6.65 5.58χ2 7.14 7.37

TABLE 13.2: Parameters that describe the magnetic structures of the Mn0.80Co0.20WO4 at2 K under 6 T field along the a axis. For simplicity the following notation has been used:k1 = ( 1

2 , 0, 0) and k2 = [-0.215(1), 12 , 0.452(2)].

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194 CHAPTER 13. MAGNETIC-FIELD-INDUCED TRANSITIONS IN MN0.80CO0.20WO4

0 4 8 12 16 200

4

8

12AF2 cycloidal? H//

Paramagnetic?

ConicalStructure

CollinealAF4

μ0H

(T)

Temperature (K)

Par

amag

netic

FIGURE 13.7: H ´ T phase diagram proposed for Mn0.80Co0.20WO4 compound withH ‖ α based on bulk magnetic measurements and polarization measurements with H ‖ α.Circles are obtained from magnetic measurements, rhombs from pulsed high field mea-surements (which has to be moved to lower temperature, as indicated by the arrows tocorrect for the magnetocaloric effect) and triangles from polarization measurements.

13.3 Summary and conclusions

Along this chapter the influence of an external field in the Mn0.80Co0.20WO4 com-position has been investigated and the results demonstrate that the stability of thedifferent phases that appear at zero field strongly depends on the direction alongwhich the field is applied.

When H ‖ b and H ‖ α + 90˝, the collinear AF4 phase does not undergo anysignificant change in fields at least up to 12 T, but on the contrary, in the conicalstructure the rotation plane of the moments flops perpendicular to the field at acritical fields that decrease with increasing temperature (from about 15 T at 2 Ktowards zero at TN2). This transition is accompanied by the disappearance of Pb.The high field magnetic structure allows electric polarization within the ac plane.

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13.3. SUMMARY AND CONCLUSIONS 195

The parameters of this high field structure and how it evolves from the one at zerofield have been determined by neutron diffraction.

If the external field is applied along the α axis, the situation is reversed: AF4becomes unstable and disappears with increasing the field and the incommensu-rate part of the cone undergoes small variations. At high fields, it is likely that theconical structure becomes a pure cycloidal structure with the electric polarizationalong b.

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196 CHAPTER 13. MAGNETIC-FIELD-INDUCED TRANSITIONS IN MN0.80CO0.20WO4

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Part IV

Summary and Conclusions

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Summary and Conclusions

To summarize, throughout the thesis several aspects of the Mn1´xCoxWO4 family(x =0, 0.05, 0.10, 0.15 and 0.20) have been investigated:

(i) Crystalline structure: both the effect of the doping and the influence of themagnetic ordering on the crystal structure has been explored. A very weakspin-lattice coupling detectible only by Larmor diffraction and high reso-lution X-ray powder diffraction, was documented for three representativeMn1´xCoxWO4 compositions, x = 0, x = 0.05 and x = 0.20. The evolutionof the cell parameters across the magnetic transitions of the latter two com-pounds is different, as their magnetic structures are. The monoclinic angle β

decreases monotonously in the x = 0.05 composition that does not undergoany change in the orientation plane where spins are confined. On the con-trary, in x = 0.20, the monoclinic angle has a minimum close to the multifer-roic transition, below the transition β starts increasing. This coincides withthe appearance of the conical spin arrangement. On the other hand, the unit-cell volume decreases with increasing the Co concentration, with no changein the orientation of the octahedra.

(ii) The zero field magnetic structures of all the compositions have been deter-mined, and therefore a complete description of the structures has been given.Their corresponding electric ordering has being provided as well.

(iii) The detailed study of the magnetic structures allowed us to observe big

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200

temperature-induced-variations in the characteristics of an ellipse (ε and am-plitudes of the semi-axes) within temperature ranges of about 10 K.

(iv) Co2+ (strongly anisotropic ion) substitution for Mn2+ (an isotropic ion) above10% results in anisotropic paramagnetic susceptibility, reflecting in differentparamagnetic temperatures along different directions and also suggests anorbital contribution to the effective magnetic moments.

(v) 15 % of cobalt is enough to orient the easy-magnetic axis close to the pureCoWO4’s one. At this doping the AF4 phase (proper of CoWO4) is alreadyvery stable, however at low temperatures competing phases emerge and co-exist due to high degree of magnetic frustration. This shows that at x = 0.15concentration the effect of the single ion anisotropy of the cobalt is slightlystronger than the rest of the interactions. It could be considered as the bound-ary concentration that separates two regions: x ă 0.15, where the Co2+ singleion anisotropy competes with the Mn2+ one; x ą 0.15 where the effect of thecobalt is dominant in the system.

(vi) Superspacial formalism1 has been applied to re-examine the incommensu-rate magnetic structures of the pure MnWO4. The study let us conclude thatthe manganese atoms within the crystallographic unit cell are related in theAF3 phase, whereas in the AF2 phase they become independent. Conse-quently the polarization arises along b. In the case of x = 0.10, in the multi-ferroic phase, both manganese atoms are still related by a mirror plane, andthis yields to electric polarization in the ac plane.

This formalism allowed us to propose a new magnetic model for describingthe AF3 phase: as an alternative to the widely accepted collinear and sinu-soidally modulated structure, a cycloidal magnetic structure is put forward.Though, both manganese atoms have different chirality, which results in op-posite electric polarization. This model corresponds to an antiferroelecticone.

(vii) Evidence for coexistence of the nonpolar-collinear commensurate magneticphase, AF1, and the polar cycloidal structure, AF2, during the field-inducedAF2-to-AF1 transition in MnWO4 is provided based on incremental mag-netic susceptibility results. These data also suggest that a trace of AF2 do-mains exist at temperature lower than the AF2-AF1 boundary of the zero

1The application of superspace formalism to the analysis of incommensurate magnetic phases is apromising development with yet few examples in the literature.

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201

magnetic field phase diagram (this observation could be regarded as a sup-port for existence of AF2 embryos being responsible for the recently publishedpolarization memory effect).

(viii) We have determined the high field X-phase of the pure MnWO4: a cycloidalstructure within the ac plane that gives rise to electric polarization within theac plane. In analogy to the Mn0.90Co0.10WO4, the X phase has been namedas AF2’. An abrupt transition has been proposed to describe the AF2-to-AF2’ transition: a increase of the eccentricity upon increasing field until therotation plane of the spins flops perpendicular to the field.

(ix) The effect of a magnetic field along the b axis on Mn0.95Co0.05WO4 has beeninvestigated: The multiferroic AF2 phase with spin-cycloid structure in theαb plane and with polarization along b undergoes a continuos transition fromthe αb plane to the wα plane, being w = α + 90˝ the hard direction2. Acomplete phase diagram is proposed based on the results from magnetic,polarization and neutron diffraction studies. Bulk magnetic measurementsand the evolution of certain magnetic reflections clearly demonstrate a field-induced transition with H ‖ α, though neutron-diffraction experiments werenot conclusive about the spin arrangement.

(x) The multiferroic AF2’ of Mn0.90Co0.10WO4 remains stable in magnetic fieldas high as 15 T applied along the b axis, while in field along c and a axes thespin-cycloid structure flops from the ac plane to directions almost perpen-dicular to the magnetic field at „ 3 T and „ 8.5 T, respectively. Those spinreorientations are accompanied by a dramatic suppression of both Pa and Pc

components of the polarization.

(xi) The stability of the distinct phases that appear in Mn0.80Co0.20WO4 at zerofield strongly depends on the direction along which the field is applied.

When H ‖ b and H ‖ α + 90˝, the collinear AF4 phase does not undergo anysignificant change in fields at least up to 12 T. On the contrary, in the conicalstructure the rotation plane of the moments flops perpendicular to the fieldat a critical fields. The critical field decreases with increasing temperature(from about 15 T at 2 K towards zero at TN2). This transition is accompaniedby the disappearance of Pb. The high field magnetic structure allows electricpolarization within the ac plane.

2This evolution of the plane was suggested by Liang et. al. [129] but our neutron diffraction studyunambiguously confirmed it.

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202

If the external field is applied along the α axis, the situation is reversed: AF4becomes unstable and disappears with increasing the field and the incom-mensurate part of the cone undergoes small variations. At high fields, it islikely that the conical structure becomes a pure cycloidal structure with theelectric polarization along b.

(xii) Magnetic phase diagrams for fields along different directions are proposedbased on the results from magnetic, polarization and neutron-diffraction stud-ies.

When writing the thesis, several studies on the topic has been advanced orcompleted, but those results are not included in the thesis. I would like to mentionthe magnetic studies up to very high fields of up to 60 T, which have allowed us totrack all the field-induced transitions that take place before the field finally forcesthe moments to align parallel to it, and thus to construct the complete magneticphase diagrams for two of the compositions.

During the thesis we have fulfilled most of our purposes, however new ques-tions have emerged that should be addressed by further investigations:

(i) What is the AF5 magnetic structure? Is it equal to AF2’?

(ii) Is the AF3 phase really an antiferroelectric one?

(iii) What underlying in the Mn0.90Co0.10WO4 to have a completely different sym-metry and consequently different properties?

(iv) What is the origin of the magnetic-history effect on zero-field propagation-vector the observed in Mn0.90Co0.10WO4?

(v) What magnetic structures are induced in Mn0.95Co0.05WO4 when the mag-netic field is applied along α?

(vi) What are the magnetic structures at high field and low temperature inMn0.80Co0.20WO4 either with field along α or b axis?

So, finding the answer to those questions could be the continuation of thework in a near future.

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Part V

Appendices

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Appendix AAppendix A: How to perform an

experiment on D23

The steps to follow to perform an experiment on the sigle-crystal neutrondiffractometer D23 with unpolarized neutrons are explained in this ap-pendix.

Actually, the main task is to orient the sample on the instrument. Once thisis done, the experiment can run and luckily, there is no need to touch the sampleagain until the end of it. Nevertheless, orientating a crystal on D23 can be a difficulttask, due to the fact that it has to be done manually and that the reflections arecollected individually.

The D23 control program is called MAD, like on many other diffractomme-ters at the ILL.

(i) Cell parameters of the crystal must be known. This information is given toMAD. That way, and knowing the wavelength of the neutron beam, the 2θ ofeach hkl reflection is known.

(ii) If the orientation of the main axes relative to the faces of the crystal areroughly known, work becomes easier, or at least one. For that, usually aLaue diffractogram is done before. Let us assume that we know one face ofthe crystal, the [010] face, the crystal will be located on the sample table withthis face parallel to the neutron beam this way the b axis is perpendicular tothe beam. If this information is lacking, the procedure becomes more compli-

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216 APPENDIX A. APPENDIX A: HOW TO PERFORM AN EXPERIMENT ON D23

cated: one has to make some hypothesis about the orientation of the crystaland go on with the orienting process.

(iii) Glue the crystal with plasticine on a goniometric head.

(iv) Once the crystal is mounted in the instrument with the b axis vertical, weknow that all (h0l) reflections are located in the ν = 0 plane, i.e. in the equa-torial plane. Therefore, we search for those reflections. For making this taskeasier, and if the crystal structure is known, one can calculate the structurefactors for all the reflections and look for the most intense ones.

(v) Look for a h0l reflection (for instance (002)): first, tell MAD to put the detectorat the 2θ position that corresponds to the (h10l1) = (002) reflection, we makesure that ν = 0 and we scan ω until we observe a peak on the rate meter.When it is found it must be centered (done automatically). If there is notintensity found, one should move ν and repeat the procedure (this meansthat the b axis is not completely vertical). Command list:

˛ hkl1 0 0 2 Detector positioned at the corresponding 2θ.

˛ pnu 0 ν = 0˝.

˛ pom 180 ω = 180˝.

˛ pom -180 ω = ´180˝.

When the peak is detected, stop scan and center the reflection from currentposition.

˛ cen4 0 0 2 Reflection centered from current position and it is giventhe (002) name.

The latter command produces a new file called centra.dat with optimizedvalues for the angles. If the former procedure is not successful, move ν andrepeat the process. Remember that γ is still well positioned.

˛ pnu 2 ν = 2˝.

˛ pom 180 ω = 180˝.

˛ pom -180 ω = ´180˝.

(vi) Then, center the Friedel reflection (00-2). Since the angular position of the(002) reflection is already known and the Friedel pair must be at the same2θ, γ(002) = γ(00´2), ω(002) ˘ 180 = ω(00´2) and ´ν(002) = ν(00´2). Hence,

˛ mom 180

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217

˛ pnu -x x is the value of ν for (002).

˛ cen4 0 0 -2

This step is not mandatory. In fact, it is helpful to correct both the height andthe precession of the sample.

(vii) Once a Friedel-pair is found, another reflection is needed. It is recommendedto find an orthogonal reflection. Let us choose (200) reflection. After center-ing (200) and (-200) reflections, one will finish with a centra.dat file thatlooks like the following:

0.0 0.0 2.0 0.0 29.673 -171.428 1.165 ...

2.0 0.0 0.0 0.0 30.864 -81.817 -0.414 ...

0.0 0.0 -2.0 0.0 29.754 8.611 -1.219 ...

-2.0 0.0 0.0 0.0 30.764 98.134 0.417 ...

where the information corresponds to h, k, l, χ, γ, ω and ν.

(viii) Thus, we end up with four reflections with their corresponding angles. Upto now, the names of the reflections were given without paying to much at-tention to that task. To make sure they are well indexed, one can verify itusing the subroutine indexd23. With the cell parameters, wavelength andthe motor positions of each reflection, it can index the reflections and pro-pose which couple of indexes satisfy the observed angles. It is important todo so, even more in our situation where the structure is monoclinic, β ą 90˝and β˚ ă 90˝, because the indexation may be wrong, see Fig. A.1.

(200)(-200)

(00-2)

(002)

(200)(-200)

(00-2)

(002)(a) (b)

𝜷* 𝜷*

FIGURE A.1: Two possible indexations: (a) the correct one and (b) the wrong one.

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218 APPENDIX A. APPENDIX A: HOW TO PERFORM AN EXPERIMENT ON D23

(ix) Then, one should run the program rafin to produce a UB matrix that relatesthe laboratory framework (XYZ) and the crystallographic axes of the crystal.The very first time that we run this subroutine none of the parameters shouldbe refined. We are just interested in a raw UB matrix, since there are notenough reflections collected to refine anything. Commands:

˛ rafin 0 0 0 0 0 0 0 0 0 0 0 0 It runs the subroutine and does notmake any refinement.

˛ getub˛ y Confirms that you want to download the produced UB matrix.

X Y Za˚ ´0.20599 ´0.02624 ´0.00246b˚ 0.00293 ´0.00672 ´0.17372c˚ 0.02171 ´0.19877 0.00805

(x) According to the previous UB matrix, b axis is almost parallel to Z (3rd col-umn) and the sample could be glued in a proper sample holder using anappropriate glue (depending on the characteristics of the material and ex-perimental conditions). It is very unlikely that the manual alignment is sogood from the beginning. Usually there is some misalignment between the baxis and Z, which are reflected in the optimized angles of the Friedel pairs inthe centra.dat file:

(a) ν = ν1+ν22 : ν ă 0 or ν ą 0 when the crystal is then too low or too high,

respectively.

(b) γ(hkl) � γ(´h´k´l) the sample is off centered. To fix this, one has toreadjust the translations of the goniometer in the plane.

(c)

when all the necessary corrections are done, the centering is performed againand a new and better UB matrix is obtained. Finally the sample is glued onan aluminum pin which will be fixed in the required sample environment.

(xi) Since the sample has been taken out from the instrument, it has to be re-oriented, but this time we have a UB matrix to start with. Hence, the proce-dure is re-started from the beginning. Command list:

˛ hkl0 0 0 2 Detector positioned at the corresponding 2θ according tothe UB matrix.

˛ pnu 0 ν = 0˝.

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219

˛ pom 180 ω = 180˝.

˛ mom -180 Move omega from current position to ω = ´180˝.

Again, when the intensity peak is detected, stop scan and center the reflectionfrom current position.

˛ initialize Re-write centra.dat file.

˛ cen4 0 0 2 Reflection centered from current position and it is given(002) name.

(xii) Find other reflections and obtain the preliminary UB matrix. Note that ifthe sample is mounted in a cryostat of a magnet, and if it is off-centeredwith respect to the beam, the complete sample-table has to be moved and theprogram XYZ calculates how much.

(xiii) To find a good and definitive UB matrix one should center about 20 or 30reflections and refine the UB (cell parameters or wavelength). First preparea file of nuclear reflections and name it, nucl.hkl for example. Try to in-clude all type of reflections that can be reached and do not restrict to (h0l), ifpossible. Command list:

˛ cen5 nucl.hkl Reflections in the file nucl.hklwill be centred.

˛ rafin 0 0 0 0 0 0 1 1 1 0 1 0 a, b, c and β will be refined (if allcentered reflections are of the type (h0l), b can not be refined).

˛ getub˛ y

The experiment is ready to be started. Now the experimental conditions needto be fixed (temperature, field. . . ). To collect data to determine a magnetic structurethere are certain steps that can be followed.

(i) As for the nuclear structure where one needs to know the cell parameters tostart the experiment, to study a magnetic structure one must know roughlythe propagation vector k of the magnetic structure. To increase the accuracyin the value of the propagation vector, it is required to have a good UB matrixwith cell parameters refined at the temperature at which the magnetic struc-ture will be measured1. Then, (hkl) ˘ k magnetic reflections can be centeredand from the obtained angles for each reflection the propagation vector canbe refined, which results in k = (kx ˘ δkx, ky ˘ δky, kz ˘ δkz).

1It is not mandatory to refine the parameters if we know that the do not vary.

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220 APPENDIX A. APPENDIX A: HOW TO PERFORM AN EXPERIMENT ON D23

(ii) Create a file with all reachable magnetic reflections (magn.hkl) and collectthem. Do the same for nuclear reflections, nucl.hkl. The latter will pro-vide the scale factor that, in turn will give the absolute value of the orderedmagnetic moment in the magnetic refinement.

˛ par sps 30000 Monitor counts.

˛ mes magn.hkl Measure reflections in file magn.hkl

˛ mes nucl.hkl Measure reflections in file nucl.hkl

(iii) It might be interesting to collect nuclear data also in the paramagnetic re-gion to see if there is any magneoelastic coupling (a change in the nuclearstructure in the magnetically ordered state).

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Appendix BRepresentation analysis

Representation Analysis can restrict the number of possible magnetic struc-tures of the material. Once the modulation wave vector is known, k, therelation between atoms in different unit cells is known, and the number of

possible magnetic structures reduced. In the following lines we will show how toobtain the possible magnetic models that are compatible with the crystal structuresymmetry. These are the steps to be followed:

(i) Find the little co-group Gk.

Gk is formed by the rotational part of the symmetry operations of the nuclearspace group that leave invariant k.

Rk + K = k (B.1)

where K is a reciprocal lattice vector and tR|tu = g P G.1

Being G = P2/c = tE, 2y, I, cu the nuclear space group and the propaga-tion vectors k1 = (˘ 1

4 , 12 , 1

2 ) and k2 = (´0.214, 12 , 0.457), depending on the

magnetic phase, in both cases Gk= m (mirror plane).

(ii) Find the little group: Gk.

The little group is a space group formed by the symmetry operations of Gk

including the translations. So, if we add the translations to Gk= m = tE, mu

we end up withGk = Pc = tE, cu (B.2)

1Symmetry operations, g, are represented as tR|tu where R is the rotational part and t the transla-tional part.

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222 APPENDIX B. REPRESENTATION ANALYSIS

(iii) Find the character table of the little co-group.

Then, the character table of the little group can be obtained using the follow-ing equation. (From now on the procedure will be detailed for k = k1).

Γkν(tRi|tiu) = e´2iπk¨tγν(tRiu) (B.3)

where Γ are the representations.

m # 1 mA’ Γ1 1 1A” Γ2 1 -1

Pc # 1 cA’ Γ1 1 -iA” Γ2 1 i

TABLE B.1: Upper table: character table of the point group m. Lower table: charactertable of the space group Pc.

(iv) Apply the symmetry operations on the position of the magnetic atoms andon the magnetic moments.

The magnetic atoms Mn1 and Mn2 sit on ( 12 , y, 1

4 ) and ( 12 , ´y, 3

2 ). The effect ofthe symmetry operations of the little group on the position of the magneticatoms are the ones shown in Table B.2.

E c1 1 22 2 1

χperm 2 0

TABLE B.2: The effect of the symmetry operations of the space group Pc on the positionsof the magnetic atoms. χperm is the character corresponding to the interchange of the sitesof the atoms, the trace of the matrix that represents the effect of the symmetry operation.

In order to see the effect of the symmetry operations on the magnetic momentof each atom, m, one can use the next equation, where mx, my and mz are the

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223

x, y and z components of the magnetic moment.

RM = Det(R)Γ(R)

⎛⎜⎝

mx

my

mz

⎞⎟⎠ (B.4)

So,

EM = 1

⎛⎜⎝

1 0 00 1 00 0 1

⎞⎟⎠

⎛⎜⎝

mx

my

mz

⎞⎟⎠ =

⎛⎜⎝

mx

my

mz

⎞⎟⎠ χaxial(E) = 3

myM = ´1

⎛⎜⎝

1 0 00 ´1 00 0 1

⎞⎟⎠

⎛⎜⎝

mx

my

mz

⎞⎟⎠ =

⎛⎜⎝

´mx

my

´mz

⎞⎟⎠ χaxial(my) = ´1

χaxial are the characters of the symmetry elements acting on the axial vectors(magnetic moments).

So, combining the effect of the symmetry operations on the sites and on themoments the total effect is shown on Table B.3.

E my

m1x: m1

x ´m2x

m1y: m1

y m2y

m1z : m1

z ´m2z

m2x: m2

x ´m1x

m2y: m2

y m1y

m2z : m2

z ´m1z

TABLE B.3: The total effect of the symmetry operations on the positions and magneticmoments of the magnetic atoms. The notation: matom

component.

(v) Fill the character table and decompose Γ into irreducible representations.

Γ = Γperm b Γaxial (B.5)

Using the following equations the decomposition can be done.

Γ =ÿ

nνΓν (B.6)

nν =1

n(Gk)

ÿg

χΓ(g)χΓν(g) (B.7)

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224 APPENDIX B. REPRESENTATION ANALYSIS

Pc 1 cΓ1 1 -iΓ2 1 i

χperm 2 0χaxial 3 i

χ 6 0

TABLE B.4: The complete character table of the space group Pc and the permutationaland axial part of the reducible representations. χ = χperm ˆ χaxial .

where n(Gk) is the order of the space group, that is the number of elementsin that group.

Γ = (Γ1 ‘ Γ2) b (Γ1 ‘ 2Γ2) = 3(Γ1 ‘ Γ2) (B.8)

There are three different basis vector for each irreducible representation thathave to be found. The linear combinations of them describe a possible mag-netic structure.

(vi) Find the basis vectors. The basis vectors are

Vαk,ν =

ÿg

Γk,ν(g)g ¨ m1k,α (B.9)

where g are the elements of Pc in this case, ν corresponds to each representa-tion and, α = x, y, z.

Γ1: Γ2:Vx

1 = m1x + im2

x = Ax Vx2 = m1

x ´ im2x = Bx

Vy1 = m1

y ´ im2y = By Vy

2 = m1y + im2

y = Ay

Vz1 = m1

z + im2z = Az Vz

2 = m1z ´ im2

z = Bz

A general magnetic structure associated with Γ1 (Γ2) can be c1Ax + c2By +

c3Az (c11Bx + c1

2Ay + c13Bz). We can conclude that in our crystal the relation between

the moments of the atoms Mn1 and Mn2 is (the result is the same for k2 but in thiscase the phase instead of being e´ πi

2 is e´πikz ):

m1x = ¯e´πikz m2

x m1y = ˘e´πikz m2

y m1z = ¯e´πikz m2

z

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Appendix CConstrains of the magnetic

modulation due to magneticsuperspace symmetry

In what follows there is a summary of the steps necessary to deduce the con-straints in the magnetic modulations introduced by the symmetry elementswithin the superspace formalism.

The information needed to start the analysis:

(i) Paramagnetic space-group: P2/c11 where 11 refers to the time inversion. Thisspace group consists of all the symmetry operations of the ordinary P2/cspace group, plus an equal number of operations obtained by multiplyingall of them by time reversal {11|0 0 0 0}

(ii) Magnetic atom: Mn1 located in the 2 f Wyckoff.

˛ Mn1: ( 12 y 1

4 )

˛ Mn2: ( 12 -y+1 3

4 )

(iii) Propagation vector: The incommensurate propagation vector, k = (´0.214, 12 , 0.457),

lies on the symmetry line G (α 12 γ) of the Brillouin Zone.

From these information one can deduce the compatible superspace groups:P2/c11(α 1

2 γ)00s and P2/c11(α 12 γ)0ss, as explained in Chapter 7.

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226APPENDIX C. CONSTRAINS OF THE MAGNETIC MODULATION DUE TO MAGNETIC SUPERSPACE

SYMMETRY

The set of generators for each magnetic superspace group can be written asfollows:

(i) mG1 : P2/c11(α 12 γ)00s

˛ {my|0 0 12 0} The glide plane does not invert the magnetic moment and

hence, the character for this operator is 1. Within this irrep, there is noneed to make a phase shift to recover the initial configuration (t4 = 0).

˛ {1|0 0 0 0}

˛ {11|0 0 0 12 }

(ii) mG2 : P2/c11(α 12 γ)0ss. In this case the plane inverts the moment, the charac-

ter for this operator is -1. Within this irrep, a phase shift has to be added tothe plane operation.

˛ {my|0 0 12

12 } In this case the glide plane does invert the magnetic mo-

ment, that is why a phase shift must be applied to recover the initialspin configuration, i. e. to obtain an undistinguishable state.

˛ {1|0 0 0 0}

˛ {11|0 0 0 12 }

C.1 P2/c11(α12γ)0ss

In this sections all the constraints imposed by the symmetry of this superspacegroup will be deduced step by step.

The symmetry cards of the symmetry operators of this superspace group(obtained from equation (2.31)) are the following:

(i) {2y|0 0 12

12 }: -x1 x2 -x3+1

2 x2-x4+12 m [HR = (0 1 0)].

Remember the effect of the two fold axis on k:

2y ¨ k = 2y ¨ (α12

γ)

= (´α12

´ γ)

= ´(α12

γ) + (010) (C.1)

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C.1. P2/C11(α 12 γ)0SS 227

The Rs matrix can be constructed by comparing equations (2.28) and(C.1).

Rs =

⎛⎜⎜⎜⎜⎝

´1 0 0 00 1 0 00 0 ´1 00 1 0 ´1

⎞⎟⎟⎟⎟⎠ (C.2)

and the general translation vector for the two fold axis is ts = (0012

12 ) in this

case.

The same procedure should be followed to obtain the symmetry cards of therest of operators.

(ii) {1|0 0 0 0}: -x1 -x2 -x3 -x4 m [HR = (0 0 0)]

(iii) {my|0 0 12

12 }: x1 -x2 x3+1

2 -x2+x4+12 m [HR = (0 -1 0)]

(iv) {1’|0 0 0 12 }: x1 x2 x3 x4+1

2 -m [HR = (0 0 0)]

Once we know the symmetry cards of the superspace group, we should seewhat are the relationships between the modulation functions of the magnetic mo-ments. Lets start with the restrains on the spin modulations of Mn1 because of 2y

site symmetry (which keeps the atomic position of Mn1 invariant):

M1(´x4 +12+ y1) = 2y ¨ M1(x4)

M1α(´x4 +12+ y1) = M1α(x4) (α = y) (C.3)

M1α(´x4 +12+ y1) = ´M1α(x4) (α = x, z) (C.4)

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228APPENDIX C. CONSTRAINS OF THE MAGNETIC MODULATION DUE TO MAGNETIC SUPERSPACE

SYMMETRY

If we write the later explicitly:

α = y

Cs1y cos(2π[´x4 +

12+ y1] + 2πφx4

1y) = Cs1y cos(2πx4 + 2πφx4

1y)

Cs1y cos(2π[x4 ´ 1

2´ y1] ´ 2πφx4

1y) = Cs1y cos(2πx4 + 2πφx4

1y)

φx41y = ´φx4

1y ´ y1 ´ 12

φx41y = ´ y1

2´ 1

4(C.5)

α = x, z

Cs1α cos(2π[´x4 +

12+ y1] + 2πφx4

1α) = ´Cs1α cos(2πx4 + 2πφx4

1α)

Cs1α cos(2π[x4 ´ 1

2´ y1] ´ 2πφx4

1α) = Cs1α cos(2π[x4 ´ 1

2] + 2πφx4

1α)

φx41α´ = ´φx4

1α ´ y1

φx41α = ´ y1

2(C.6)

Mn1 and Mn2 are related by the inversion center, so lets see what are therestrictions forced by this relation. {I|0 0 0 0}Mn1 Ñ Mn2 :

M2(´x4) = M1(x4) ñ equal amplitude and (C.7)

cos(2π[´x4 + φx42α]) = cos(2π[x4 + φx4

1α])

φx42α = ´φx4

1α (α = x, y, z). (C.8)

To summarize, the P2/c11(α 12 γ)0ss superspace group restrains the magnetic

modulations in the following way:

M1x(x4) = M01x cos[2π(x4 ´ y1

2)] (C.9)

M1z(x4) = M01z cos[2π(x4 ´ y1

2)] (C.10)

M1y(x4) = M01y cos[2π(x4 ´ y1

2´ 1

2)] = M0

1y sin[2π(x4 ´ y1

2)] (C.11)

M2x(x4) = M01x cos[2π(x4 +

y1

2)] (C.12)

M2z(x4) = M01z cos[2π(x4 +

y1

2)] (C.13)

M2y(x4) = ´M01y sin[2π(x4 +

y1

2)] (C.14)

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C.2. P2/C11(α 12 γ)00S 229

There are 3 independent parameters to refine: M01x, M0

1y and M01z. The refine-

ment has been done using JANA2006 [61] program. The program does not use therestrictions as in equations (C.9)-(C.14) directly, but it develops those cosines andsines in cosines and sines that are just functions of x4. So the relation between theequations (C.9)-(C.14) and the parameters that JANA2006 refines is:

M01x cos[2π(x4 ´ y1

2)]

= M01x[cos(2πx4) cos(2π

y1

2) ´ sin(2πx4) sin(2π

y1

2)] (C.15)

M01x cos(2π

y1

2) = Mc

1x (C.16)

M01x sin(2π

y1

2) = Ms

1x (C.17)

C.2 P2/c11(α12γ)00s

We will see now what are the constraints imposed by the P2/c11(α 12 γ)00s super-

space group. The symmetry cards of the operators in this superspace group arelisted below:

(i) {2y|0 0 12 0}: -x1 x2 -x3+1

2 x2-x4 m [HR = (0 1 0)].

The Rs matrix is equal to the one of the two fold axis in the previous spacegroup, but the translational part, ts differs.

Rs =

⎛⎜⎜⎜⎜⎝

´1 0 0 0 00 1 0 0 00 0 ´1 0 1

20 1 0 ´1 0

⎞⎟⎟⎟⎟⎠ (C.18)

The rest of the operations are:

(ii) {1|0 0 0 0}: -x1 -x2 -x3 -x4 m [HR = (0 0 0)]

(iii) {my|0 0 12

12 }: x1 -x2 x3+1

2 -x2+x4 m [HR = (0 -1 0)]

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230APPENDIX C. CONSTRAINS OF THE MAGNETIC MODULATION DUE TO MAGNETIC SUPERSPACE

SYMMETRY

The constraint in the modulations due to the two fold axis are:

M1(´x4 + y1) = 2y ¨ M1(x4)

M1α(´x4 + y1) = M1α(x4) (α = y) (C.19)

M1α(´x4 + y1) = ´M1α(x4) (α = x, z) (C.20)

If we write the later explicitly:

α = y

Cs1y cos(2π[´x4 + y1] + 2πφx4

1y) = Cs1y cos(2πx4 + 2πφx4

1y)

cos(2π[x4 ´ y1] ´ 2πφx41y) = cos(2πx4 + 2πφx4

1y)

φx41y = ´φx4

1y ´ y1

φx41y = ´ y1

2(C.21)

α = x, z

Cs1α cos(2π[´x4 + y1] + 2πφx4

1α) = ´Cs1α cos(2πx4 + 2πφx4

1α)

Cs1α cos(2π[x4 ´ y1] ´ 2πφx4

1α) = Cs1α cos(2π[x4 +

12] + 2πφx4

1α)

φx41α = ´φx4

1α ´ y1 ´ 12

φx41α = ´ y1

2´ 1

4(C.22)

In this also case the inversion center relates the Mn1 and Mn2 as it did in theprevious case, see (C.7) and (C.8). So, the modulation functions can be expressedas follows:

M1x(x4) = M01x cos[2π(x4 ´ y1

2´ 1

4)] = M0

1x sin[2π(x4 ´ y1

2)] (C.23)

M1z(x4) = M01z sin[2π(x4 ´ y1

2)] (C.24)

M1y(x4) = M01y cos[2π(x4 ´ y1

2)] (C.25)

M2x(x4) = M01x cos[2π(x4 +

y1

2+

14)] = ´M0

1x sin[2π(x4 +y1

2)] (C.26)

M2z(x4) = ´M01z sin[2π(x4 +

y1

2)] (C.27)

M2y(x4) = M01y cos[2π(x4 +

y1

2)]. (C.28)

The number of independent parameters in this case is the same, 3.

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C.3. P211(α 12 γ)0S 231

C.3 P211(α12γ)0s

In Chapter 7 we have seen that the superspace group P211(α 12 γ)0s in the primitive

lattice describes properly the magnetic structure of the multiferroic AF2 phase.Now we will derive the constraints forced by this symmetry.

The symmetry operators that are contained in this group and their symmetrycards are listed below:

(i) P211(α 12 γ)0s :

˛ {2y|0 0 12

12 } : -x1 x2 -x3+1

2 x2-x4+12 m

˛ {1’|0 0 0 12 } : x1 x2 x3 x4+1

2 -m

Since Mn1 and Mn2 are related by the glide plane and this element is lost, thetwo Mn atoms in the crystallographic unit cell become independent. So, a priorithere will be six parameters to refine (three per atom). However the magneticmodulation functions for each atom are constrained by the 2y symmetry operationin the same way that they were in P2/c11(α 1

2 γ)0ss.

Mi(´x4 +12+ yi) = 2y ¨ Mi(x4)

Miα(´x4 +12+ yi) = Miα(x4) (α = y) (C.29)

Miα(´x4 +12+ yi) = ´Miα(x4) (α = x, z) (C.30)

where the index i refers to Mn1 and Mn2. So, from this we can conclude that

φx4iy = ´ yi

2´ 1

4(C.31)

φx4iα = ´ yi

2(α = x, z) (C.32)

and the modulation functions can be described as

M1x(x4) = M01x cos[2π(x4 ´ y1

2)] (C.33)

M1z(x4) = M01z cos[2π(x4 ´ y1

2)] (C.34)

M1y(x4) = M01y sin[2π(x4 ´ y1

2)] (C.35)

M2x(x4) = M02x cos[2π(x4 ´ y2

2)] (C.36)

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232APPENDIX C. CONSTRAINS OF THE MAGNETIC MODULATION DUE TO MAGNETIC SUPERSPACE

SYMMETRY

M2z(x4) = M02z cos[2π(x4 ´ y2

2)] (C.37)

M2y(x4) = ´M02y sin[2π(x4 ´ y2

2)]. (C.38)

C.4 Non-conventional centering X

The conditions of the modulations can be simplified if the propagation vector (α 12

β) is changed to (α 0 β) by using a double unit cell along b. The component 12 of the

propagation vector only means that the magnetic moment, M, of atoms related bylattice translations (0 1 0) are in anti phase:

M at cell (0 0 0) = ´M at cell (0 1 0)). (C.39)

This can be described more simply by using a cell: a, 2b, c such that now wehave four Mn atoms per unit cell; but we include in the symmetry a centering (0 1

20 1

2 ). If

Mn1 : (12

y1

214) Mn2 : (

12

´ y1

234) (C.40)

Mn11 : (12

y1 + 12

14) Mn22 : (

12

´ y1 + 12

34) (C.41)

the modulations of the atoms Mn11 and Mn22 are determined by those of Mn1 andMn2:

MMn11(x4) = MMn1(x4 +12) (C.42)

MMn22(x4) = MMn2(x4 +12) (C.43)

The symmetry group of AF3 can then be described as X2/c11(α0γ)0ss withsymmetry operators:

(i) {2y|0 0 12

12 }

(ii) {1|0 0 0 0}

(iii) {my|0 0 12

12 }

(iv) {1’|0 0 0 12 }

and the X meaning the centering (0 12 0 1

2 ). The modulation function are nowreduced to either cosine or sine functions:

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C.4. NON-CONVENTIONAL CENTERING X 233

(i) {2y|0 0 12

12 }:

M1(´x4 +12) = 2y ¨ M1(x4) (C.44)

M1x(x4) = Mc1x cos(2πx4) (C.45)

M1z(x4) = Mc1z cos(2πx4) (C.46)

M1y(x4) = Ms1y sin(2πx4) (C.47)

(C.48)

The change of origin along x4 can make {2y|0 0 12 0}, and sine and cosine are

interchanged.

(ii) {1|0 0 0 0}:

M2(´x4) = ´M1(x4) (C.49)

M2x(x4) = ´M1x(x4) = Mc1x cos(2πx4) (C.50)

M2z(x4) = ´M1y(x4) = Mc1z cos(2πx4) (C.51)

M2y(x4) = ´M1z(x4) = Ms1y sin(2πx4) (C.52)

Check: For the previous description, the x4 of an atom Mn1 was: x4 = q ¨ (R -rν) = kx

2 + kz4 + y1

2 + k ¨ R, in the new description, x’4 = kx2 + qz

4 + k ¨ R, hence,cos(2πx4 ´ y1

2 ) = cos(2πx14).

P211(α 12 γ)0s becomes X211(α0γ)0s and the conditions restricting Mn1(x4)

and Mn2(x4) are maintained but their interrelations disappear.

Page 252: Magnetic order, spin-induced ferroelectricity and ... · Magnetic order, spin-induced ferroelectricity and structural studies in multiferroic Mn 1´xCo xWO 4 by IRENE URCELAY OLABARRIA

234APPENDIX C. CONSTRAINS OF THE MAGNETIC MODULATION DUE TO MAGNETIC SUPERSPACE

SYMMETRY

Page 253: Magnetic order, spin-induced ferroelectricity and ... · Magnetic order, spin-induced ferroelectricity and structural studies in multiferroic Mn 1´xCo xWO 4 by IRENE URCELAY OLABARRIA

Appendix DHow to intersect the space groups

associated to mG1 and mG2

In the case of the symmetry operations transforming k into ´k, the translationalpart along the coordinate x4 depends on the choice of the origin in the internalspace, i. e. it depends on the global phase associated with the modulation. In or-der to derive the symmetry of the superposition of two active irreps, one must thenexplicitly consider this dependence. When there is a single irrep incommensuratemodulation, one is always allowed to choose this phase as zero. However, if twoprimary irrep modulations are superposed, only one of the modulation phases canbe arbitrarily chosen by fixing an origin, and the relative phase difference betweenthe two modulations becomes physically relevant. Consequently, the global su-perspace symmetry depends in general on the relative phase shift of the two irrepmagnetic modulations.

If an incommensurate system has a generic symmetry operation tR, θ|t, τ0u, ashift of the global phase of the modulation by a quantity φ (in 2π units), is equiva-lent to a translation of the origin of the internal coordinate x4 by ´φ. Under this ori-gin shift, the above symmetry operation becomes tR, θ|t, τ0 ´ RIφ + φu, where RI

is defined in equation (2.28). This means that the operations that keep k invariantdo not change, while those transforming k into ´k transform into tR, θ|t, τ0 + 2φu.In our case, the inversion center and the two fold axis transform k into ´k, but theplane does not. Hence, the glide plane must be unchanged, whereas the inversioncenter and the two fold axis should be transformed.

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236 APPENDIX D. HOW TO INTERSECT THE SPACE GROUPS ASSOCIATED TO MG1 AND MG2

˛ mG1: X2/c11(α 12 γ)00s:

– {my|0 0 12 0}

– {1|0 0 0 2φ}

– {2y|0 0 12 2φ}

– {1|01201

2 }

˛ mG2: X2/c11(α 12 γ)0ss.

– {my|0 0 12

12 }

– {1|0 0 0 2φ1}

– {2y|0 0 12

12+2φ1}

– {1|01201

2 }

The intersection of the symmetry groups for the different primary irrep modesdepends on their global phases. Let us start by calculating the operations derivefrom different φ values.

(i) mG1: φ = 0, 12

˛ {my|0 0 12 0}

˛ {1|0 0 0 0}

˛ {2y|0 0 12 0}

˛ {1|01201

2 }

(ii) mG1: φ = 14 , 3

4

˛ {my|0 0 12 0}

˛ {1|0 0 0 12 }

˛ {2y|0 0 12

12 }

˛ {1|01201

2 }

(iii) mG1: φ = arbitrary

˛ {my|0 0 12 0}

˛ {1|0 0 0 2φ}

˛ {2y|0 0 12 2φ}

Page 255: Magnetic order, spin-induced ferroelectricity and ... · Magnetic order, spin-induced ferroelectricity and structural studies in multiferroic Mn 1´xCo xWO 4 by IRENE URCELAY OLABARRIA

237

˛ {1|01201

2 }

(iv) mG2: φ1 = 0, 12

˛ {my|0 0 12

12 }

˛ {1|0 0 0 0}

˛ {2y|0 0 12

12 }

˛ {1|01201

2 }

(v) mG2: φ1 = 14 , 3

4

˛ {my|0 0 12

12 }

˛ {1|0 0 0 12 }

˛ {2y|0 0 12

32 }

˛ {1|01201

2 }

(vi) mG2: φ1 = arbitrary

˛ {my|0 0 12

12 }

˛ {1|0 0 0 2φ1}˛ {2y|0 0 1

212+2φ1}

˛ {1|01201

2 }

So, now two irrep modes have to be intersected. Remember that both mag-netic modes may have the same symmetry and therefore, one has to intersect amGi mode with another mGi mode or a mGj mode. The phase of the former willbe fixed to zero.

Page 256: Magnetic order, spin-induced ferroelectricity and ... · Magnetic order, spin-induced ferroelectricity and structural studies in multiferroic Mn 1´xCo xWO 4 by IRENE URCELAY OLABARRIA

238 APPENDIX D. HOW TO INTERSECT THE SPACE GROUPS ASSOCIATED TO MG1 AND MG2

Page 257: Magnetic order, spin-induced ferroelectricity and ... · Magnetic order, spin-induced ferroelectricity and structural studies in multiferroic Mn 1´xCo xWO 4 by IRENE URCELAY OLABARRIA

Appendix ESymmetry of the AF2’ multiferroic

phase of Mn0.90Co0.10WO4

In the case of the 10% doped compound the polarization arises in the ac plane.Therefore, we should first try the Xc11(α0γ)ss group, which emerges from theintersection of the mG2 mode that describes AF3 phase with another mode

of the same symmetry but phase shifted, with Δφ arbitrary. In order to know therestrictions imposed on the magnetic modulations by this superspace group, wewill repeat the procedure as for the pure compound. The symmetry operations ofthis group can be expressed as:

(i) {my|0 0 12

12 } : x1 -x2 x3+1

2 x4+12 m

(ii) {1’|0 0 0 12 } : x1 x2 x3 x4+1

2 -m

In this case, Mn1 and Mn2 are related by glide plane. So, both atoms are notindependent.

M2(´x4 +12) = ´myM1(x4)

α = y

φx41y + φx4

2y = ´12

(E.1)

α = x, z

φx41α + φx4

2α = 0 (E.2)

Page 258: Magnetic order, spin-induced ferroelectricity and ... · Magnetic order, spin-induced ferroelectricity and structural studies in multiferroic Mn 1´xCo xWO 4 by IRENE URCELAY OLABARRIA

240 APPENDIX E. SYMMETRY OF THE AF2’ MULTIFERROIC PHASE OF MN0.90CO0.10WO4

and the modulation functions can be described as

M1x(x4) = M01x cos[2π(x4 + φ1x)] (E.3)

M1z(x4) = M01z cos[2π(x4 + φ1z)] (E.4)

M1y(x4) = M01y cos[2π(x4 + φ1y)] (E.5)

M2x(x4) = M01x cos[2π(x4 + ´φ1x)] (E.6)

M2z(x4) = M01z cos[2π(x4 ´ φ1z)] (E.7)

M2y(x4) = M01y cos[2π(x4 ´ φ1y ´ 1

2)] = M0

1y sin[2π(x4 ´ φ1y)]. (E.8)

As there are no operations in the superspace group with RI = ´1 (change x4

to ´x4), one of the phases can be chosen arbitrarily. The parameters that describethe magnetic structure at 2 K have been obtained from the refinement of 273 inde-pendent reflections averaged from 482 reflections with Rint = 2.24. The agreementfactors and the parameters are summarized in Table E.1.

Temperature Ms1x Ms

1y Ms1z Mc

1x Mc1y Mc

z1

2 K -2.33(9) -0.08(7) -3.28(8) 0.00** 0.00(6) -2.33(8)RF = 5.17%, RwF = 5.99% and RwF2 = 9.95%

** Fixed manually.

TABLE E.1: Refined parameters of the multiferroic phase of the pure Mn0.90Co0.10WO4

2 K in the superspace group Xc11(α0γ)ss.

Page 259: Magnetic order, spin-induced ferroelectricity and ... · Magnetic order, spin-induced ferroelectricity and structural studies in multiferroic Mn 1´xCo xWO 4 by IRENE URCELAY OLABARRIA

Publications

Directly related to the thesis

(i) I. Urcelay-Olabarria, E. Ressouche, A. A. Mukhin, V. Yu. Ivanov, A. M. Bal-bashov, G. P. Vorob’ev, Yu. F. Popov, A. M. Kadomtseva, J. L. García-Muñoz,and V. Skumryev.“Neutron diffraction, magnetic and magnetoelectric studies of phase transitions inMn0.90Co0.10WO4 multiferroic.”Phys. Rev. B 85, 094436 (2012).

(ii) I. Urcelay-Olabarria, E. Ressouche, A. A. Mukhin, V. Yu. Ivanov, A. M. Bal-bashov, J. L. García-Muñoz, and V. Skumryev.“Conical antiferromagnetic order in the ferroelectric phase of Mn0.80Co0.20WO4 re-sulting from the competition between collinear and cycloidal structures.”Phys. Rev. B 85, 224419 (2012).

(iii) I. Urcelay-Olabarria, J. L. García-Muñoz, E. Ressouche, V. Skumryev, V. Yu.Ivanov, A. A. Mukhin and A. M. Balbashov.“Lattice anomalies at the ferroelectric and magnetic transitions in cycloidal Mn0.95Co0.05WO4

and Mn0.80Co0.20WO4 multiferroics.”Phys. Rev. B 86, 184412 (2012).

(iv) I. Urcelay-Olabarria, J. M. Perez-Mato, J. L. Ribeiro, J. L. García-Muñoz, E.Ressouche, V. Skumryev and A. A. Mukhin.“Incommensurate magnetic structures of multiferroic MnWO4 studied within thesuperspace formalism.”accepted in Phys. Rev. B.

Page 260: Magnetic order, spin-induced ferroelectricity and ... · Magnetic order, spin-induced ferroelectricity and structural studies in multiferroic Mn 1´xCo xWO 4 by IRENE URCELAY OLABARRIA

242 APPENDIX E. SYMMETRY OF THE AF2’ MULTIFERROIC PHASE OF MN0.90CO0.10WO4

(v) I. Urcelay-Olabarria, E. Ressouche, A. A. Mukhin, V. Yu. Ivanov, A. M.Kadomtseva, Yu. F. Popov, and G. P. Vorob’ev, A. M. Balbashov, J. L. García-Muñoz and V. Skumryev.“Evolution of MnWO4 magnetic structure and electric polarization across phasetransition induced by magnetic field along b axis.”to be submitted to Phys. Rev. B.

Other publications during the PhD period

(i) A. Faik, E. Iturbe-Zabalo, I. Urcelay and J. M. Igartua.“Crystal structures and high-temperature phase transitions of the new ordered dou-ble perovskites Sr2SmSbO6 and Sr2LaSbO6.”Solid State Chemistry 182 (2009) 2656-2663.

(ii) A. Faik, I. Urcelay, E. Iturbe-Zabalo and J. M. Igartua.“Cationic ordering and role of the A-site cation on the structure of the new doubleperovskites Ca2´xSrxRSbO6 (R = La, Sm) (x = 0, 0.5, 1).”J. of Molecular Structure 963 (2010) 145-152.

(iii) T.I. Milenov, P.M. Rafailov, I. Urcelay-Olabarria, E. Ressouche, J. L. García-Muñoz, V. Skumryev, M. M. Gospodinov.“Magnetic behavior of La2CoMnO6´δ crystal doped with Pb and Pt.”Materials Research Bulletin 47 (2012) 4001Ð4005.